Radicals and Anti-radicals of near-rings and Matrix near-rings. 1 Anthony Mpho Matlala Under the supervision of Prof. J. F. T. Hartney and Prof. B. A. Watson School of Mathematics University of the Witwatersrand, Johannesburg Private Bag 3, P.O. WITS 2050, South Africa March 3, 2008 1Submitted to the University of the Witwatersrand, Johannesburg, in fulfill- ment of the requirements for the degree of Doctor of Philosophy. 1 Declaration I declare that this thesis is my own, unaided work. It is being submitted for the Degree of Doctor of Philosophy in the University of the Witwater- srand, Johannesburg. It has not been submitted before for any degree or examination in any other university. Anthony Mpho Matlala This day of , at Johannesburg, South Africa. 2 Abstract In this thesis the ideal structure of near-rings and the associated matrix near- rings is studied. Firstly, two aspects of near-rings are investigated, these are: the decomposition of a near-ring; and the nilpotency of the Jacobson s- radical, via the nil-rigid series. Secondly, the interplay of ideals of a near-ring and the ideals of the associated matrix near-ring is investigated. The socle ideal (an anti-radical) of a near-ring R is characterized as an intersection of annihilators of R-groups of a special type, called type-K. That is, the socle ideal is an annihilator ideal. The R-groups of type-K are shown to separate the Jacobson 0-radical and the Jacobson s-radical, if there is at least one R-group of type-K contained in some type-0 R-group. Alternating chains of R-groups of type-0 and type-K are defined and will be called ?-chains. It is then proved that the maximal length of the ?-chains give a lower bound on the nil-rigid length of the near-ring. Consequently, one can directly determine that the nil-rigid length of the matrix near-ring, Mn(R), is equal or bigger than the nil-rigid length of R from the structure 3 of any one of the finite faithful R-groups of R. For an ideal I being a Jacobson type radical, socle ideal or s-socle ideal of a near-ring, open-questions on the relationship between I+, I? and the matrix near-ring ideals of the same type are answered. A systematic theory is developed that connects each annihilator ideal I of R to its corresponding matrix near-ring ideal, I. This is made possible by the use of Action 2 of the matrix near-ring on the Mn(R)-groups. Specific conditions are given which ensure that I+ ? I and I ? I?. Examples illustrate cases in which no such relationships exist. Necessary and sufficient conditions are given for I+ ? I ? I? to hold. This thesis thus illustrates the relationship between the structure of a faithful R-group and the ideal structure of the associated matrix near-rings. 1 Acknowledgements I thank Prof. John Hartney for introducing me to the study of near-rings. I am so ever grateful to Prof. Bruce Watson. It was through his support, motivation and encouragement that I had the strength to complete this dissertation. I thank all my colleagues, especially those I had regular interactions with, for a pleasant atmosphere in the school. Special thanks to all who were toiling with their PhD studies, while I was doing the same, for a studious environment and sometimes a reminder that I am not alone in this journey. Many thanks to Alley Manthata and Josh Mdawe for providing me with the basic necessities of life, ?My Bros, keep doing the same to others.? I am deeply indebted to my brother, Baile, who is younger but has been like a big brother through tough times. Thanks to my sister, Tsholofelo, for such constant love that she has lavished on me. I thank God for my kids, Kelebone and Neo, they are a constant reminder of what true love is. Thanks be to God Almighty for Charles and Kgomotso Matlala, the best parents anyone could wish for. The talents in me, the person I am, the perseverance, the rescilience, all come from both my parents. The thought of the sacrifices they made, just to see me through school, boggles my mind. Contents 1 Introduction 4 2 Preliminaries 13 2.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Jacobson type radicals . . . . . . . . . . . . . . . . . . . . . 18 2.3 Anti-radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Matrix near-rings . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Some Characterizations 36 3.1 A characterization of Soi(R) . . . . . . . . . . . . . . . . . . . 37 3.1.1 Type-K R-groups and the Jacobson s-radical . . . . . 39 3.1.2 A classification of monogenic R-groups . . . . . . . . . 44 3.2 A characterization of the nil-rigid series . . . . . . . . . . . . . 53 3.2.1 The class K(?) and the nil-rigid length . . . . . . . . . 53 3.2.2 The ?-chains . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.3 ?-ideal series and the nil-rigid series . . . . . . . . . . 60 3.3 Examples on ?-chains . . . . . . . . . . . . . . . . . . . . . . 69 4 Annihilator Ideals in Mn(R) 73 2 CONTENTS 3 4.1 Modules in matrix near-rings . . . . . . . . . . . . . . . . . . 74 4.2 Intersections of annihilators . . . . . . . . . . . . . . . . . . . 82 4.3 The ideals Soi(R)+ and Soi(R)? . . . . . . . . . . . . . . . 90 4.4 Examples on Socle Ideals . . . . . . . . . . . . . . . . . . . . . 94 5 ?-Primitivity in Mn(R) 101 5.1 ?-Radicals of Mn(R) . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Examples on ?-radicals . . . . . . . . . . . . . . . . . . . . . . 106 5.3 The s-socles . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4 Examples on s-socles . . . . . . . . . . . . . . . . . . . . . . . 116 6 Nil-rigid series of Mn(R) 127 6.1 The nil-rigid length of Mn(R) . . . . . . . . . . . . . . . . . . 128 6.2 Series of ideals of Mn(R) . . . . . . . . . . . . . . . . . . . . . 132 6.3 Examples on the nil-rigid length of Mn(R) . . . . . . . . . . . 138 7 Conclusion 143 7.1 A brief summary of results . . . . . . . . . . . . . . . . . . . . 143 7.2 Some Open Questions . . . . . . . . . . . . . . . . . . . . . . 146 A Action of Mn(R) on factor-groups 149 B Structure of ideals of the form I+ 153 Chapter 1 Introduction In this thesis the ideal structure and specific decompositions of zero symmet- ric near-rings, and their associated matrix near-rings, are studied. For this, a natural point of departure is the radical theory of near-rings. There is a large literature on the analogues of various ring radicals in the near-ring context. These results have been collected in, the now standard, text-books on near-rings, by Pilz [23] and Meldrum [14]. Radicals of near-rings differ remarkably from their analogous ring radicals. For instance, it is well-known, [1] and [12], that there exist finite distributively generated near-rings whose prime ideals are not maximal. 4 CHAPTER 1. INTRODUCTION 5 The focus of this thesis is the connection between Jacobson type radicals and an anti-radical called the socle ideal. Unlike in the ring case, for near- rings there are at least three distinct Jacobson type radicals which are two- sided ideals, viz. J?(R), (? = 0, s, 2). Substantial work has been done on the Jacobson 0-radical, J0(R), and the Jacobson 2-radical, J2(R). Special attention is therefore given here to the Jacobson s-radical, Js(R), introduced by Hartney in [5]. The Jacobson s-radical contains the Jacobson 0-radical, and is itself contained in the Jacobson 2-radical, that is, J0(R) ? Js(R) ? J2(R). The s-radical is crucial to understanding the structure of near-rings and is fundamental in this thesis. Another ideal which plays an integral part in elucidating the structure of near-rings, introduced by Hartney, in [5], is the socle ideal of a near-ring. The socle ideal, denoted Soi(R), is referred to as an anti-radical as it annihilates one of the Jacobson type radicals, namely the Jacobson 0-radical. It is known that a zero-symmetric near-ring, R, with identity and satisfying the descending chain condition on left ideals, can be decomposed as R = Soi(R)? L, where L is a left ideal of R containing J0(R), see Hartney [5]. Hartney, in [5], showed that there is a duality between the socle ideal and the Jacobson s-radical, given by Js(R) = (0) if and only if Soi(R) = R. CHAPTER 1. INTRODUCTION 6 The intricate relation between the socle ideal and the Jacobson s-radical is beautifully displayed by the nil-rigid series of a near-ring. The nil-rigid series, first defined by Scott in [25], is an ascending alternating series of Jacobson 0-radicals and socle ideals of quotient near-rings. The length of the nil-rigid series of a near-ring can be used as a measure of how far the Jacobson s-radical is from being nilpotent, see [6]. It is well known, [23] and [14], that Jacobson type radicals can be char- acterized as intersections of annihilators of monogenic R-groups of type-? (? = 0, s, 2). It is proved, in this thesis, that the socle ideal can also be char- acterized as an intersection of annihilators of monogenic R-groups of yet another type, referred to here as type-K. The ideal structure of a near-ring is thus seen to be dictated to by the structure of its faithful R-groups. The structure of a near-ring, R, is further revealed by considering alternating chains, called ?-chains, of monogenic R-subgroups of types 0 and K of any faithful R-group. The lengths of the ?-chains are related to the length of the nil-rigid series and in some cases determine it. The structure of near-rings can be further unravelled by considering the ideal structure of their matrix near-rings. In near-rings, the lack of one distributive law results in complications when matrices with entries from a near-ring are considered. For only particu- lar classes of near-rings do the matrices over near-rings form a near-ring, called a matrix near-ring, under the usual matrix addition and multiplica- CHAPTER 1. INTRODUCTION 7 tion. Heatherly [10], and Ligh [13], studied such classes of near-rings and the associated matrix near-rings. In 1986, Meldrum and Van der Walt took a different approach to form matrix near-rings from an arbitrary near-ring. They regarded an n ? n matrix, over a near-ring, R, as a function from the group Rn to itself. For a near-ring which is a ring, its matrix near-ring constructed this way coincide with its matrix ring. Here we shall use the definition of a matrix near-ring as given by Meldrum and Van der Walt, [18], denoted by Mn(R). For an ideal I of a near-ring R, Meldrum and Van der Walt [18], introduced two ways to construct an ideal in Mn(R), namely I+ and I?, with I+ ? I?. In addition, Van der Walt [27], showed that the inclusion can be strict. The gap between these two ideals, I+ and I?, can be wide, in that there may exist chains of ideals between them, see Meyer [21]. He refers to these ideals of Mn(R) as intermediate ideals. In a later paper, [17], Meldrum and Meyer furnished an example showing that the Jacobson 0-radical of a matrix near- ring might be an intermediate ideal. A substantial amount of work has been done in studying this interplay of ideals of a near-ring R and corresponding ideals of Mn(R). In particular, it is known that (J2(R)) + ? J2(Mn(R)) = (J2(R))?, proved by Van der Walt in [26]. The theory of s-primitivity in matrix near- rings, by Hartney and Mavhungu, [7, 9], showed that Js(Mn(R)) ? (Js(R))?. CHAPTER 1. INTRODUCTION 8 That (Js(R)) + ? Js(Mn(R)), is proved in this thesis, published in [8]. In his doctoral thesis, [19], Meyer proved that J0(Mn(R)) ? (J0(R))?. The relationship between (J0(R))+ and J0(Mn(R)) has been an open ques- tion. Meldrum and Meyer, [17], conjectured that, for zero symmetric near- rings, (J0(R))+ ? J0(Mn(R)). However, counter examples to this conjec- ture exist as is shown in this thesis. For socle ideals, as with the Jacobson 0-radicals, it is shown, in this thesis, that there is in general no relationship between (Soi(R))+ and Soi(Mn(R)). On the other hand, we prove that Soi(Mn(R)) ? (Soi(R))?. The s-socle of a near-ring R is the minimal ideal modulo which Js(R) is non-zero nilpotent. One would thus expect the s-socles to have relationships resembling those of the Jacobson type radicals and those of the socle ideals. Surprisingly, there is in general no relationship between the s-socle of Mn(R), denoted A, and the ideal A?, where A is the s-socle of R. Examples of near-rings are given where A? ? A, and where A? and A not comparable. Yet, for zero symmetric near-rings satisfying the descending chain condition on R-subgroups (DCCS), it is proved that A+ ? A . CHAPTER 1. INTRODUCTION 9 Although, we are mainly interested in ideals which are intersections of anni- hilators of monogenic R-groups, monogenic R-groups are not the only R- groups needed to understand the ideal structure of matrix near-rings. Mel- drum and Meyer, in [15], showed that, for a locally monogenic R-group, ?, which is not monogenic, its corresponding Mn(R)-group, ?n, is monogenic under Action 2. Non-monogenic R-groups, ?, such that ?n is monogenic as an Mn(R)-group, are classified into three classes, those of 0n-form, sn- form and Kn-form. These R-groups play a key role in understanding the relationship between the ideals of R and those of Mn(R). For example, Meldrum and Meyer, [15], used them to show that it is possible to have J0(Mn(R)) negationslash= J2(Mn(R)), even though J0(R) = J2(R). In [9], Hartney and Mavhungu showed that, in this example by Meldrum and Meyer, J0(Mn(R)) = Js(Mn(R)) negationslash= J2(Mn(R)), while J0(R) = Js(R) = J2(R). To further illustrate the effect of these non-monogenic R-groups on the ideals of Mn(R), an example of a near-ring R is given here, for which J0(Mn(R)) negationslash= Js(Mn(R)) negationslash= J2(Mn(R)), with J0(R) = Js(R) = J2(R). CHAPTER 1. INTRODUCTION 10 The existence of non-monogenic R-groups of 0n-form and Kn-form affects the lengths of the ?-chains, and hence the length of the nil-rigid series of R. In general, a lower bound for the length of the nil-rigid series of Mn(R) can be determined from the lengths of the ?-chains. Deeper studies into group-representation theory should yield a better understanding of the ideal structure of near-rings. This thesis is structured as follows. In Chapter 2, definitions of basic concepts and preliminary results on near- rings and their associated matrix near-rings are collected. Chapter 3 presents a characterization of the socle ideal as an intersection of annihilators of monogenic R-groups of type-K and gives a description of the nil-rigid series in terms of annihilators of monogenic R-groups which are members of chains of R-groups, called ?-chains. A series of ideals, constructed from the ?-chains, is defined and shown to contain the nil-rigid series. This new series of ideals is called the ?-ideal series. Examples are given to illustrate how the maximal length of ?-chains is related to the nil-rigid length. In Chapter 4, the connection between annihilator ideals of a near-ring and those of the associated matrix near-rings is studied. By an annihilator ideal we are referring to any intersection of annihilating ideals of monogenic R- groups (or Mn(R)-groups). The results are aided by the recognition of some non-monogenic R-groups whose corresponding matrix near-ring groups are CHAPTER 1. INTRODUCTION 11 monogenic. This theory is immediately applied to the socle ideal, as an example of an annihilator ideal. In Chapter 5, some open-questions on ?-primitive ideals of matrix near-rings are answered. In addition, questions on the relationship between the s-socle ideal of a near-ring and the s-socle ideal of the associated matrix near-ring are answered. Examples of near-rings are given illustrating cases where no such relationships exist. Chapter 6 presents an application of a combination of the theory of ?-chains, from Chapter 3, and the theory of annihilator ideals, from Chapter 4, to the nil-rigid series of matrix near-rings. As a consequence, one can give a lower bound on the nil-rigid length of the matrix near-ring from the ?-chains. Examples of near-rings, with their nil-rigid lengths strictly less than those of their associated matrix near-rings, are given. Throughout this thesis, R denotes a zero symmetric right near-ring with identity, and Mn(R) the Meldrum-Van der Walt matrix near-ring associated with R. Both R and Mn(R) are assumed to satisfy the DCCS, unless otherwise stated. The term R-group is used instead of R-module. Likewise, the term R- kernel is used instead of R-ideal. An ideal will always mean a two-sided ideal. A faithful R-group of R will usually be denoted by ?. A direct-sum of n copies of an R-group, ?, will be denoted by ?n, where n is a natural number. The Greek letter ? is reserved to denote the nil-rigid length. To CHAPTER 1. INTRODUCTION 12 make a distiction between the nil-rigid length of R and Mn(R), we use ?R and ?Mn(R), respectively. Chapter 2 Preliminaries In this chapter we collect definitions of basic concepts and preliminary results needed in this thesis. Concepts of a near-ring, an R-group and an ideal of a near-ring are defined in Section 2.1. In Section 2.2, ?-primitive ideals are defined and the relevant results on ?-radical theory are collected. Section 2.3 introduces anti-radicals: the socle ideal; the crux; and the critical ideal. Relevant known theory on how these ideals relate is recorded here. Meldrum- Van der Walt matrix near-rings are defined in Section 2.4. Basic facts on the ideal theory of matrix near-rings are collected here. 13 CHAPTER 2. PRELIMINARIES 14 2.1 Basic concepts In this section we collect basic definitions and introduce the notation we will use throughout this thesis. Definition 2.1.1 A near-ring (R,+, ?) is a set with two binary operations, satisfying the following conditions: (i) (R,+) is a group, written additively (not necessarily abelian); (ii) (R, ?) is a semi-group; (iii) (a+ b) ? c = a ? c+ b ? c , for all a, b, c ? R. A near-ring, such as defined above, is called a right distributive near-ring (or right near-ring). If (iii) is replaced by the left distributive rule, the resulting near-ring is referred to as a left near-ring. For right distributive near-rings yx = (y + 0)x = yx+ 0x, for all x, y ? R. Hence, 0x = 0, for all x ? R. However, it need not be the case that x0 = 0, for all x ? R. If the condition x0 = 0, for all x ? R, holds on the near-ring R, then R is said to be a zero-symmetric near-ring. CHAPTER 2. PRELIMINARIES 15 A typical example of a zero-symmetric near-ring is a set of zero respecting mappings from a group, ?, to itself, i.e., M0(?) = {f : ? ? ? | f(0) = 0 }, under pointwise addition and composition of maps. Throughout this thesis, M0(?) denotes the near-ring of zero respecting map- pings from a group, ?, to itself, and R denotes a right near-ring. Definition 2.1.2 Let (?,+) be a group (not necessarily abelian) and let R be a right near-ring with identity. Let ? be a map ? : R? ? ? ? and denote ?(r, ?) =: r?, for r ? R,? ? ?. Then (?, ?) is called an R-group if for r1, r2 ? R, ? ? ?, (i) (r1 + r2)? = r1? + r2?, (ii) (r1r2)? = r1(r2?), (iii) 1? = ?. We shall denote the R-group (?, ?) simply by ? if the meaning of ? is clear. A subgroup (?,+) of (?,+) is called an R-subgroup of (?, ?) if (?, ?|R??) is itself an R-group, in this case we will say that ? is an R- subgroup of ?. An R-group ? is said to be monogenic if there exists a ? in ? such that ? = R?. Since R is a right near-ring, then an R-group is a near-ring analogue of a left-module of a ring. For each right near-ring R with identity, the group (R,+) is also an R-group under left-multiplication. CHAPTER 2. PRELIMINARIES 16 Definition 2.1.3 Let R be a right near-ring with identity, and let ? be an R-group. An R-subgroup, ?, of ?, is an R-kernel of ? if (i) (?,+) is a normal subgroup of (?,+), (ii) r(? + ?)? r? ? ?, for all r ? R, ? ? ?, and ? ? ?. Definition 2.1.4 Let R be a right near-ring with identity, ? be an R- group, and let L be an R-kernel of ?, and ?/L = {? + L | ? ? ?} denote the quotient group ?/L. Then (?/L,+) is called the factor R-group of ? by L, where R acts on ?/L by r(? + L) := r? + L, for all ? ? ?, r ? R. Note that if the R-group, ?, in Definition 2.1.4, is monogenic, then the quotient group, ?/L, is a monogenic R-group. Throughout this thesis, for any R-group ?, the expression ? = ? ??? ?? will always mean that ? is a near-ring direct sum of the R-kernels ??, ? ? ?. Definition 2.1.5 Let R and R? be right near-rings with identity, and ? and ? be R-groups. Then (i) ? : R ? R? is a (near-ring) homomorphism if ?(r1 + r2) = ?(r1) + ?(r2) and ?(r1r2) = ?(r1)?(r2), CHAPTER 2. PRELIMINARIES 17 for all r1, r2 ? R. (ii) ? : ? ? ? is an R-homomorphism if ?(? + ??) = ?(?) + ?(??) and ?(r?) = r?(?), for all ?, ?? ? ?, r ? R. The ideals of the near-ring R may be considered as the kernels of (near-ring) homomorphisms of R. Definition 2.1.6 Let R be a right near-ring. A normal subgroup (I,+) of (R,+) is an ideal of R if (i) IR ? I, (ii) r(r? + ?)? rr? ? I, for all r, r? ? R, ? ? I. If (ii) holds then I is called a left ideal of R. Likewise I is called a right ideal if (i) holds. The left ideals of R are precisely the kernels of R-homomorphisms of (R,+). Quotient near-rings are defined in the usual sense. Definition 2.1.7 Let R be a right near-ring and let I be an ideal of R. Then the (additive) factor group R/I is a near-ring under the binary oper- ations: (r + I) + (s+ I) = (r + s) + I, r, s ? R and (r + I)(s+ I) = (rs) + I, r, s ? R. CHAPTER 2. PRELIMINARIES 18 This near-ring is called the quotient near-ring of R by I. The annihilating ideal of an R-group ? is defined to be (0 : ?) := {x ? R | x? = 0, ?? ? ? }. An R-group ? is a faithful R-group if (0 : ?) = (0). Proposition 2.1.8 [23, Proposition 1.48] Let R be a zero-symmetric right near-ring. If ? is a faithful R-group, then R ?? M0(?). That is R has an embedding in M0(?). Hence every zero-symmetric right near-ring can be considered as a subnear- ring of M0(?), for some group (?,+). If the near-ring R satisfies the descending chain condition for R-subgroups, we say R satisfies the DCCS. Similarly, if R satisfies the descending chain condition for left ideals, we say R satisfies the DCCL. 2.2 Jacobson type radicals In this section we define the ?-radicals, ? = 0, 12 , s, 2, and recall the known radical theory needed in this thesis. CHAPTER 2. PRELIMINARIES 19 Definition 2.2.1 Let R be a right near-ring with identity. A monogenic R-group, ? = R?, for some ? ? ?, is of (i) type-0 if it has no non-trivial R-kernels, (ii) type-s if it is of type-0 and for all ?? ? ? with R?? negationslash= (0) we have that there exists a near-ring direct sum decomposition, R?? = k? i=1 ?i, where each ?i is an R-kernel of R?? and an R-group of type-0, (iii) type-2 if it has no non-trivial R-subgroups. It is clear from Definition 2.2.1 that type-2 R-groups are of type-s, hence of type-0. Lemma 2.2.2 [5] Let R be a right near-ring and ? an R-group. If ? = ? ??? ??, a direct sum of R-groups ??, ? ? ?, all of type-0, then any non-trivial R-kernel, H, of ?, is a direct summand of ? with a direct sum of some of the ???s as a co-summand. Theorem 2.2.3 [7, Theorem 2.1] Let R be a right near-ring satisfying the DCCL and let ? be a faithful R-group. Any R-group of type-0 is a homomorphic image of some monogenic R-subgroup R? of ?. The following proposition due to Betsch (see Pilz, [23, Proposition 3.14]), gives that for an R-group, ?, to be an (R/I)-group, where R/I is the quotient near-ring of R by I, it is necessary and sufficient that I annihilates ?. CHAPTER 2. PRELIMINARIES 20 Proposition 2.2.4 Let R be a right near-ring, I be an ideal of R, and ? a group and ? ? {0, s, 2}. (a) If ? is an R-group with I ? (0 : ?) and we define (r + I)? := r?, then ? is an (R/I)-group under this operation. If ? is an R-group of type- ?, then it is an (R/I)-group of type-?. If ? is a faithful R-group, then it is faithful as an (R/I)-group. (b) If ? is an (R/I)-group and we define r? := (r + I)?, then ? is an R-group under this operation with I ? (0 : ?)R. If ? is an (R/I)-group of type-?, then it is of type-? as an R-group. If ? is a faithful (R/I)-group, then I = (0 : ?)R. We note that for any two R-groups, ?1 and ?2, ?1 ? ?2 ? (0 : ?1) ? (0 : ?2). Corollary 2.2.5 Let R be a zero-symmetric right near-ring, and let ?1 and ?2 be R-groups. If ?1 ? ?2, then ?2 is an (R/I)-group, where I = (0 : ?1). Proposition 2.2.6 Let R be a zero-symmetric right near-ring with identity, I be an ideal of R, and ? any R-group. If (0 : ?) ? I then (0 : ?) R/I = (0 : ?)/I, CHAPTER 2. PRELIMINARIES 21 where (0 : ?) is the annihilating ideal of ? in R, and (0 : ?)R/I is the annihilating ideal of ? in R/I. Proof The identification, (r + I)? = r?, of Proposition 2.2.4, yields the equality as follows, (0 : ?) R/I = {x+ I ? R/I | (x+ I)? = 0, ? ? ? ? } = {x+ I ? R/I | (x+ I)? = x? = 0, ? ? ?} = {x+ I ? R/I |x ? (0 : ?) } = (0 : ?)/I. square Definition 2.2.7 Let R be a right near-ring, and ? be an R-group. (i) An ideal I of R is a ?-primitive ideal if I = (0 : ?) = {x ? R | x? = 0,?? ? ? }, where ? is of type-?, ? = 0, s, 2. (ii) For ? = 0, s, 2, the Jacobson-type radical, J?(R), is the intersection of all ?-primitive ideals. (iii) The near-ring R is ?-primitive if it possesses a faithful R-group of type-?, ? = 0, s, 2. That is, if the zero ideal of R is a ?-primitive ideal. The radicals J?(R), ? = 0, s, 2, are related as follows J0(R) ? Js(R) ? J2(R). CHAPTER 2. PRELIMINARIES 22 Definition 2.2.8 [14, page 87] Let R be a right near-ring. The left ideal L of R is a ?-modular left ideal, ? = 0, s, 2, if the factor group R/L is an R-group of type-? and R has a right identity modulo L, that is, an element e such that xe? x ? L for all x ? R. Definition 2.2.9 [14, page 93] Let R be a right near-ring. J1/2(R) = ? {L |L is a 0-modular left ideal of R }. Whereas J?(R) is a two-sided ideal of R, for each ? = 0, s, 2, J1/2(R) is, in general, only a left ideal of R. Proposition 2.2.10 [14, Theorem 5.27] Let R be a right near-ring. J0(R) is the largest two-sided ideal of R contained in J1/2(R). Proposition 2.2.11 [4] If R is a right near-ring satisfying the DCCL , then Js(R) is the smallest ideal of R containing J1/2(R). Theorem 2.2.12 [4] If R is a right near-ring satisfying the DCCL, then any ideal I of R, I negationslash= (0), which contains J1/2(R) is an intersection of s-primitive ideals of R. From Propositions 2.2.10 and 2.2.11, it follows that J0(R) ? J1/2(R) ? Js(R) ? J2(R). Proposition 2.2.13 [24] If R is a right near-ring satisfying the DCCS , then J0(R) is a nilpotent ideal of R. CHAPTER 2. PRELIMINARIES 23 For the definition of d.g. near-rings we refer the reader to [14], and for such near-rings we have the following theorem. Theorem 2.2.14 [11] Let R be a distributively generated right near-ring. Then (i) J0(R) contains all nilpotent ideals of R. (ii) J1/2(R) contains all nilpotent left ideals of R. (iii) J2(R) contains all nilpotent R-groups. Lemma 2.2.15 [11] Let R be a right near-ring satisfying the DCCS. (i) If I is an intersection of maximal left ideals, then R/I = t? i=1 ?i, where each ?i is an R-group of type-0. (ii) If I = J1/2(R), then any type-0 R-group is isomorphic to one of the ?i appearing in the direct decomposition R/I = t? i=1 ?i. 2.3 Anti-radicals In this section, three anti-radicals are defined, and their theory relevant to this thesis is presented. CHAPTER 2. PRELIMINARIES 24 Lemma 2.3.1 [2] (Left distribution over R-kernels) Let R be a right near-ring, and ? be an R-group such that ? = ? i?? ?i. Then r(?1 + ?2 + ? ? ?+ ?t) = r?1 + r?2 + ? ? ?+ r?t for all r ? R and ?i ? ?i. Definition 2.3.2 [5] Let R be a right near-ring, and let F be the collection of all ideals I of R such that I = ? i?? Rei where (i) Rei is of type-0 for each i ? ?, (ii) e2i = ei, eiej = 0 for i > j with some ordering on the index set ?. An ideal I of R is said to have an F-decomposition, or to be F- decomposable, if I ? F . If F negationslash= ?, then the unique maximal element of F is called the socle-ideal of R, denoted Soi(R). If F = ?, then Soi(R) is defined to be zero. Lemma 2.3.3 [5] If R is a right near-ring satisfying the DCCL, then Soi(R) ? J1/2(R) = (0). In particular, if R = Soi(R), then J1/2(R) = (0). It follows, therefore, that (Soi(R))(J1/2(R)) = (Soi(R))(J0(R)) = (0) and so Soi(R) is an anti-radical. CHAPTER 2. PRELIMINARIES 25 Theorem 2.3.4 [5] If R a right near-ring satisfying the DCCL, then R = Soi(R) ? L where Soi(R) = t? i=1 Rei and L = t? i=1 (0 : ei). Definition 2.3.5 Let R be a right near-ring satisfying the DCCL. The de- composition R = Soi(R)? L is called the socle-decomposition of R. Theorem 2.3.6 [5] Let R be a right near-ring satisfying the DCCS and I be an ideal of R, then I ? J1/2(R) = (0) if, and only if, I ? Soi(R). Theorem 2.3.7 [5] Let R be a near-ring with DCCL and I be an ideal of R, then Soi(R/I) = R/I if, and only if, J1/2(R) ? I. Corollary 2.3.8 [5] Js(R) is the unique smallest ideal amongst all ideals I of R such that Soi(R/I) = R/I. Thus Soi(R) = R if, and only if, Js(R) = (0). In [5], Hartney investigated the connection between Soi(R) and the Crux of R. The Crux was first defined by Scott, see [25]. An ideal I of R is a nil-ideal if every element in I is nilpotent. The nil-radical of R, nil(R), is the sum of all nil-ideals of R and is itself a nil-ideal. Definition 2.3.9 Let R be a right near-ring. An ideal I of R is said to be rigid if, whenever J is an ideal of R contained in I, then (I/J) ? CHAPTER 2. PRELIMINARIES 26 nil(R/J) = (0). The sum of all rigid ideals of R is called the Crux of R, denoted Crux(R). Because of its strong nil ?avoidance? property, Crux(R) annihilates the nil-radical, nil(R), and is thus an anti-radical. Theorem 2.3.10 [5] For each right near-ring, R, Soi(R) ? Crux(R). Concerning the nilpotence of Js(R), Hartney proved the following result. Theorem 2.3.11 [5] For the right near-ring, R = Soi(R) ? L, let W := Js(R) ? Soi(R). Then Js(R/W ) is nilpotent if, and only if, Js(R) = W ? J1/2(R). In [6], Hartney used the nil-rigid series of a near-ring R to further explore the nilpotence of Js(R). Definition 2.3.12 [6, 25] For the right near-ring R satisfying the ascending chain condition for ideals, let L1 = nil(R) and C1 be the ideal containing L1 such that C1/L1 = Crux(R/L1). Let L2 be the ideal of R containing C1 such that L2/C1 = nil(R/C1). CHAPTER 2. PRELIMINARIES 27 If ? is a non-limit ordinal, define L? to be the ideal of R containing C??1 such that L?/C??1 = nil(R/C??1), and define C? to be the ideal of R containing L? such that C?/L? = Crux(R/L?). If ? is a limit ordinal, define C? = ? ? 1. Then there exists a unique ideal A of R such that A is minimal amongst all ideals I of R for which Js(R/I) is non-zero and nilpotent. Moreover, A ? Js(R) ? C??1 and Js(R/A) = Js(R)/A. Definition 2.3.15 The unique minimal ideal A in Lemma 2.3.14 will be called the s-socle of R. Theorem 2.3.16 [6] Let R be a right near-ring satisfying the DCCS and having nil-rigid length, ? > 2. Then the s-socle of R is not contained in Soi(R). Theorem 2.3.17 [6] Let R be a right near-ring satisfying the DCCS and let ? be the nil-rigid length of R. Then the s-socle, A, of R, is contained in C1 if, and only if, ? = 2. Theorem 2.3.18 [6] Let R be a right near-ring satisfying the DCCS and having nil-rigid length, ? = 2. If Soi(R) ? J0(R) = C1, then the s-socle, A of R is contained in the Soi(R). Theorem 2.3.19 [6] Let R be a right near-ring satisfying the DCCS and having nil-rigid length, ? > 1. Then the s-radical Js(R) can be decomposed CHAPTER 2. PRELIMINARIES 29 as Js(R) = J1/2(R) + A+B, where A is the s-socle of R and B = Js(R) ? L??1. The last anti-radical to be considered is the critical ideal. This ideal was first defined by Laxton and Machin for distributively generated (d.g.) near- rings, in [12]. If ? and ? are R-groups and ? is an R-homomorphic image of an R- subgroup of ?, then we say ? is a subfactor of ?, denoted ? lessmuch ?. For each faithful R-group ?, the R-isomorphism classes of irreducible R- groups which appear as subfactors of monogenic R-subgroups of ? can be split into two distinct classes H1 and H2. Here H1 contains only R-groups of type-0 and H2 contains the rest, Laxton and Machin in [12]. The mono- genic R-subgroups of ? now can be split into two classes H1 and H2 such that (a) H1 contains those monogenic R-subgroups of ? which are direct sums of elements of H1. (b) H2 contains those monogenic R-subgroups of ? with complete series and whose subfactors are elements of H2. Definition 2.3.20 Let R be a right d.g. near-ring with identity satisfying the DCCL, and ? be a faithful R-group. The critical ideal of R is defined CHAPTER 2. PRELIMINARIES 30 by Crit(R) = ? R??H2 (0 : R?), where H2 is the class of monogenic R-subgroups of R with complete series and whose subfactors are elements of H2. It was shown in [12] that, if R is a d.g. near-ring satisfying DCCL, then for each faithful R-group ?, R = L ? Crit(R), where L is a left ideal whose irreducible subfactors are isomorphic to ele- ments of H2. Here Crit(R) = q? i=1 Rei where the Rei are minimal left ideals isomorphic to elements of H1. Theorem 2.3.21 [12, Corollary 3] Let R be a right d.g. near-ring with identity satisfying the DCCL. The maximal class, H1, of R-groups of type- 0 is the same for all faithful R-groups and consequently, H1 and Crit(R) depend only on R. Thus, the critical ideal, Crit(R), is unique up to isomorphism. Hartney, in his doctoral thesis, extended the Laxton-Machin results to near- rings which are not necessarily d.g. near-rings and showed that if R satisfies the DCCS , then Crit(R) ? Soi(R). CHAPTER 2. PRELIMINARIES 31 Here monogenic R-subgroups of a faithful R-group ?, are classified into three classes: K(?), G(?) and B(?). For a near-ring R with DCCL, the following facts are established in Chapter 3: 1. The analogue of the Critical ideal, denoted by C(?), is equal to the Socle ideal. 2. If the near-ring R decomposes into R = (Soi(R)) ? L, then every element in G(?) has an isomorphic copy in Soi(R), and every element in B(?) has an isomorphic copy in L. 2.4 Matrix near-rings In this section, we introduce matrix near-rings, as defined by Meldrum and Van der Walt. Also, basic results on the ideals of matrix near-rings are collected here. The set Mn(R) of n ? n matrices, with entries from a given near-ring R, does not usually form a near-ring under the usual matrix operations. In 1986, Meldrum and Van der Walt gave an alternative definition for a matrix near-ring, cf. [18]. They regarded an n ? n matrix over a near-ring R as a function from Rn to Rn, where Rn is the direct sum of n copies of the group (R,+). Definition 2.4.1 Let R be a right near-ring with an identity. The matrix near-ring, Mn(R), associated with a near-ring R, is the subnear-ring of CHAPTER 2. PRELIMINARIES 32 M0(Rn) generated by the set { f rij | r ? R, 1 ? i, j ? n}. Here f rij : R n ? Rn is given by f rij(??) = ?i(rpij(??)), for ?? ? Rn, where ?i : R ? Rn and pii : Rn ? R are the i-th injection and projection functions, respectively. Unlike earlier attempts to define a matrix near-ring using matrix ring opera- tions, this definition gives a matrix near-ring which is always a near-ring. If the base near-ring is a ring, then the matrix near-ring defined as in Definition 2.4.1 is a matrix ring. Lemma 2.4.2 [18] Let R be a right near-ring with an identity, and f rij be as defined in Definition 2.4.1. For any a, b, ai ? R, i = 1, 2, . . . , n, (i) faij + f b ij = f a+b ij , (ii) faij + f b kl = f b kl + f a ij, if i negationslash= k, (iii) faijf b kl = ? ? ? fabil if j = k, f 0il if j negationslash= k, (iv) faij(f a1 1k1 + f a2 2k2 + ? ? ?+ f an nkn) = f a ij f aj jkj = faajikj , (v) a is zero-symmetric in R if, and only if, faij is zero-symmetric in Mn(R), (vi) a is distributive in R if, and only if, faij is distributive in Mn(R). CHAPTER 2. PRELIMINARIES 33 It follows that Mn(R) is a right distributive, zero-symmetric near-ring with identity, 1M = f 111 + f 1 22 + ? ? ?+ f 1 nn. Definition 2.4.3 Let R be right near-ring with identity. Any matrix U in Mn(R) can be represented as an expression, X, involving only f rij?s, r ? R. The length, l(X), of an expression X is the number of f rij?s in it. The weight, w(U), of a matrix U is the length of an expression of minimal length for U . Known theory about of near-rings can therefore be applied to matrix near- rings. For instance, the Jacobson-type radicals of matrix near-rings can be defined in the usual way and are related, as in R, by J0(Mn(R)) ? J1/2(Mn(R)) ? Js(Mn(R)) ? J2(Mn(R)). The relationship between ideals of a near-ring R and the ideals of its asso- ciated matrix near-ring Mn(R) form the major part of this thesis. For an ideal I of R, there are two natural ways to construct an ideal in Mn(R) which relate naturally to I, as follows. Definition 2.4.4 [18] Let R be a right near-ring with an identity, and I be an ideal of R. Then I+ := Id { faij | a ? I, 1 ? i, j ? n } and I? := (In : Rn) = {U ?Mn(R) |U?? ? In, ? ?? ? Rn } CHAPTER 2. PRELIMINARIES 34 are ideals of Mn(R). It follows readily that I+ ? I?. The ideals I+ and I? can be distinct, as shown in [27]. In fact, chains of ideals of Mn(R) may exist between I+ and I?, as shown by Meyer in [16, 21], and are called intermediate ideals. Lemma 2.4.5 [20] Let R be a right near-ring with an identity. If L is a left ideal of R and I is the largest two-sided ideal of R contained in L, then L? = I?, where L? := {W ?Mn(R) |W?? ? Ln, ? ?? ? Rn}. It follows immediately that (J1/2(R))? = (J0(R))?. Proposition 2.4.6 [27] Let R be a right near-ring with an identity. For ideals I and J of R we have, (i) I ? J implies I+ ? J+ and I? ? J?, (ii) if I notsubseteql J , then I+ notsubseteql J?, (iii) (I ? J)? = I? ? J?, in fact, this holds for arbitrary intersections, (iv) (I ? J)+ ? I+ ? J+, (v) (I + J)? ? I? + J?, (vi) (I + J)+ = I+ + J+. Lemma 2.4.7 [19, Corollary 1.54] Let R be a zero-symmetric right near- ring with identity satisfying the DCCS and I be any ideal of R. Then I+ is nilpotent in Mn(R) if, and only if, I is nilpotent in R. Ideals of Mn(R) can be related back to ideals of R, in the following manner. CHAPTER 2. PRELIMINARIES 35 Definition 2.4.8 Let D be an ideal of Mn(R), then D? := { r ? R | r = pij(U??), for someU ? D, ?? ? R n, ? 1 ? j ? n}, is a two-sided ideal of R. Proposition 2.4.9 [27, 18] Let R be a right near-ring with identity. For all ideals I of R and J of Mn(R), we have, (i) J ? (J?)?, (ii) (I?)? = I , (iii) ((J?)?)? = J?, (iv) (J?)+ ? J . Proposition 2.4.10 (Corollary 8, [27]) Let R be a right near-ring with an identity. The ideal I of R is nilpotent if, and only if, I+ is nilpotent in Mn(R). Combining the results of [7, 9, 15, 26, 20] we have the following theorem. Theorem 2.4.11 Let R be a zero-symmetric right near-ring with identity, satisfying the DCCS. Then (i) (J2(R))+ ? J2(Mn(R)) = (J2(R))?, (ii) Js(Mn(R)) ? (Js(R))?, (iii) J0(Mn(R)) ? (J0(R))? = (J1/2(R))?. Note that in [26], part (i) was proved for a more general class of near-rings than we have considered in Theorem 2.4.11, whereas in [7], (ii) was proved for R satisfying the DCCL. Chapter 3 Some Characterizations In this chapter, firstly, we characterize the socle ideal of a near-ring as an intersection of annihilators of R-groups from a special class of monogenic R-groups. These monogenic R-groups are referred to as R-groups of type- K. Secondly, we define alternating chains of R-groups called ?-chains. We study two main questions. (a) How does the existence of R-groups of type-K relate to the existence of a gap between the 0-radical J0(R) and the s-radical Js(R)? (b) Do the lengths of the ?-chains determine the nil-rigid length of a near- ring? The socle ideal is characterized in Section 3.1, and in Section 3.2 we intro- duce ?-chains and a new series of ideals of R, called the ?-ideal series. It is shown how the ?-ideal series relates to the nil-rigid series. 36 CHAPTER 3. SOME CHARACTERIZATIONS 37 In Section 3.3, the theory developed in the first two sections is illustrated with examples. 3.1 A characterization of Soi(R) In this section, a new type of monogenic R-group is defined, and will be called type-K. Monogenic R-subgroups of a faithful R-group, ?, are classified into three classes, K(?), G(?) and B(?). It is then proved that the socle ideal is an intersection of annihilators of elements of K(?). It is further shown that every member of G(?) is isomorphic to a direct summand of the socle ideal. Throughout this section, R is assumed to be a zero symmetric right near- ring with identity satisfying the DCCL and ? is assumed to be a faithful R-group. We refer to the decomposition R = Soi(R) ? L, of Theorem 2.3.4, as the socle-decomposition of R. Here L is a left ideal such that L ? J1/2(R). For near-rings satisfying the DCCS, as noted in Theorem 2.3.6, the socle ideal is the largest ideal that annihilates J1/2(R). We extend this result to near-rings satisfying the DCCL. CHAPTER 3. SOME CHARACTERIZATIONS 38 In Theorem 3 of [11], it is proved that for a d.g. near-ring, R, satisfying the DCCS, we have the decomposition R/J1/2(R) = n? i=1 ?i, where each ?i is an R-group of type-0, for i = 1, 2, . . . , n. The fact that R is distributively generated plays no role in the proof of this result, while the chain condition was simply needed to ensure that J1/2(R) could be written as an intersection of finitely many maximal left ideals. The proof of the following lemma is identical to the proof of Theorem 3 in [11]. Lemma 3.1.1 Let L be a left ideal of the right near-ring, R, with identity. Suppose L = n? i=1 Li, an intersection of finitely many maximal left ideals Li. Then the factor R-group, R/L, can be decomposed as R/L = ?1 ??2 ? ? ? ? ??n, where ?i = R?i, for some ?i ? ?i, is of type-0, and Li is the annihilating left ideal of ?i, i = 1, 2, . . . , n. Proposition 3.1.2 Let R be a zero symmetric right near-ring with identity. If ? is a monogenic R-group with a near-ring direct sum decomposition, ? = ? ? H, then the R-subgroups, ? and H, are monogenic. Proof Let ? = R?. Since ? = ? ? H is a near-ring direct sum, each of the summands, ? and H, is an R-kernel of ?. Let the unique CHAPTER 3. SOME CHARACTERIZATIONS 39 representation of ? be ? = ? + h, where ? ? ? and h ? H. Now, let d ? ? and h? ? H, then d + h? ? ?. Since ? is monogenic, we have d + h? = r? for some r in R. By left distributivity over R-kernels, Lemma 2.3.1, d+ h? = r? = r(? + h) = r? + rh, hence ?r?+d = rh?h?. Since ? and H are R-subgroups of ?, ?r? ? ? and ?h?, rh ? H. Thus, we have ?r?+d = rh?h? ? ??H = (0), which gives d = r? and h? = rh. Since d and h are arbitrary elements in ? and H, respectively, we conclude that R? = ? and Rh = H. square 3.1.1 Type-K R-groups and the Jacobson s-radical In this subsection we show that a gap between the 0-radical and the s-radical can be attributed to the presence of an R-group of type-K contained in some R-group of type-0. The symbol ?G denotes the group theoretic direct sum. Definition 3.1.3 Let R be a right near-ring with identity. A monogenic R-group, ? = R?, for some ? ? ?, is of type-K if it is not of type-0, and it has no type-0 R-kernels as its near-ring direct summands. CHAPTER 3. SOME CHARACTERIZATIONS 40 Example 3.1.1 Let ? := Z2 ?GZ2, ?1 := Z2 ?G{0} and ?2 := {0}?GZ2. Let R := { f ? M0(?) | f(?i) ? ?i, i = 1, 2; ?? ?? ? ?2 ? f(?)? f(??) ? ?2, ?, ?? ? ?}, then R is a near-ring with a multiplicative identity under point-wise addition and map composition. The group ? is a monogenic R- group, in particular ? = R(1, 1). However, ? is not of type-0 because ?2 is its only non-trivial R-kernel. Since ?1 is not an R-kernel of ?, the type-0 R-group ?2 is not a near-ring direct summand of ?. Hence ? is an R-group of type-K as it has no non-trivial R-kernels of type-0, other than ?2. Example 3.1.2 Let ? := Z4 ?G Z2 ?G Z2, H1 := Z4 ?G {0} ?G {0}, H2 := {0}?GZ2 ?GZ2, D1 := {0, 2}?G{0}?G {0}, D2 := {0}?GZ2 ?G{0} and D3 := {0} ?G {0} ? Z2. Consider R := { f ? M0(?) | f(Hi) ? Hi, i = 1, 2; f(Dj) ? Dj, j = 1, 2, 3; ? ? ?? ? H2 ? f(?)? f(??) ? H2, ?, ?? ? ?; h1 ? h2 ? D2 ? f(h1)? f(h2) ? D2, h1, h2 ? H2 }. Then R is a near-ring with a multiplicative identity under point-wise addi- tion and map composition. We make the following observations. (a) The group ? is a monogenic R-group, in particular ? = R(3, 1, 1). (b) The subgroup H2 of ? is the only non-trivial R-kernel of ?. Thus, ? is not of type-0. (c) The subgroup D2 = R(0, 1, 0) is an R-group of type-0 and it is the only non-trivial R-kernel of H2. Hence, H2 is not of type-0. (d) The R-kernel, D2, of H2, is not a near-ring direct summand of H2, because D1 is not an R-kernel of H2. Since H2 has no other non-trivial R-kernels, it is thus of type-K. CHAPTER 3. SOME CHARACTERIZATIONS 41 (e) The subgroup H1 = R(3, 0, 0) is of type-0, but it is not an R-kernel of ?. On the other hand, H2 is an R-kernel of ? but it is not of type-0. Therefore, ? does not have a non-trivial near-ring direct sum decomposition and thus ? is an R-group of type-K. Example 3.1.3 Let R be a right near-ring with identity and ? be a mono- genic R-group with ? ?= Z4, if such exists. Now if ? has an R-kernel H ?= {0, 2} ? Z4, then ? is not of type-0. In addition, since ? ?= Z4 and its only direct sum decomposition into R-kernels is itself, Z4, and since ? is not of type-0, it follows that ? is of type-K. We now show that every type-0 R-group which is not of type-s is a monogenic R-group which has no R-kernels but contains an R-group of type-K. Lemma 3.1.4 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, then each R-group of type-0 which is not of type-s contains an R-group of type-K. Proof Let ? be an R-group of type-0 which is not of type-s, then there exists a monogenic R-subgroup ? negationslash= (0) of ?, such that each near-ring direct sum decomposition of ? has at least one direct summand which is not of type-0. Although ? is monogenic, it can not be of type-0, as this would contradict the assumption that ? is not of type-s. Thus, if ? has no near-ring direct summands of type-0, it is of type-K, and the proof is complete. CHAPTER 3. SOME CHARACTERIZATIONS 42 On the other hand, if ? has at least one near-ring direct summand of type-0, then it can be written as a near-ring direct sum decomposition, ? = ?1 ?H1, where H1 is a non-trivial type-0 R-subgroup of ?. Since ? is not of type- 0, its R-subgroup ?1 negationslash= (0), and ? supersetnoteql ?1. By Proposition 3.1.2, ?1 is monogenic. Again, ?1 can not be of type-0 as this would contradict the assumption that ? is not of type-s. If ?1 has no near-ring direct summand of type-0, it is of type-K, which completes the proof for this case. Otherwise, ?1 has a least one near-ring direct summand of type-0, and hence, can be decomposed into a near-ring direct sum, ?1 = ?2 ?H2, where H2 is a non-trivial type-0 R-subgroup of ?1. Note that ?1 supersetnoteql ?2 negationslash= (0), because ?1 is not of type-0. Also, it follows from Proposition 3.1.2 that ?2 is monogenic. But ?2 can not be of type-0, because ? is not of type-s. Continuing inductively, if ?i is not of type-K, this procedure can be re- peated, but it terminates by the DCCS at, say ?n?1 = ?n ?Hn, where ?n is a non-zero type-K R-subgroup of ?n?1, and Hn is a non-trivial type-0 R-subgroup of ?n?1. square CHAPTER 3. SOME CHARACTERIZATIONS 43 Theorem 3.1.5 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS. Then J0(R) negationslash= Js(R) if, and only if, there exists an R-group ? of type-0 with an R-subgroup ? of type-K. Proof ? Suppose J0(R) negationslash= Js(R). Then there exists an R-group of type-0 which is not of type-s, say ?. By Lemma 3.1.4, ? contains an R-group of type-K. ? Conversely, let ? be an R-group of type-0, and ? be a type-K R- subgroup of ?. By definition of type-K, the R-group ? is monogenic but not of type-0, and it has no type-0 R-kernel as a direct summand. Since ? ? ?, it follows that ? is of type-0 but not of type-s. Therefore J0(R) negationslash= Js(R). square The next result extends Proposition 2.2.4 to R-groups of type-K. Proposition 3.1.6 Let R be a zero-symmetric right near-ring with identity, and ? be an R-group. Let I be an ideal of R with I ? (0 : ?), then ? is an (R/I)-group of type-K if, and only if, ? is an R-group of type-K. Proof By Proposition 2.2.4, ? is an (R/I)-group, and for each R-subgroup ? of ?, we have I ? (0 : ?) ? (0 : ?), and ? is an (R/I)-subgroup of ?. Conversely, by Proposition 2.2.4, if ? is an (R/I)-subgroup of ?, then ? is an R-subgroup of ?. Now, ? is of type-0 as an R-group if, and only if, it is of type-0 as an (R/I)-group, by Proposition 2.2.4. Thus, ? is not of type-0 as an R-group CHAPTER 3. SOME CHARACTERIZATIONS 44 if, and only if, it is not of type-0 as an (R/I)-group. Note that ? is a (group) direct summand of ? as an R-group if, and only if, ? is a (group) direct summand of ? as an (R/I)-group. A straight forward calculation, using the identification, (r + I)? := r?, made in Proposition 2.2.4 (a), shows that (r + I)[? + ?]? (r + I)? = r(? + ?)? r?, for all ? ? ?, r ? R and ? ? ?. Thus, ? is an R-kernel of ? if, and only if, ? is an (R/I)-kernel of ?. Hence, as an R-group, ? has no near-ring direct summand of type-0 if, and only if, as an (R/I)-group, it has no near-ring direct summand of type-0. square 3.1.2 A classification of monogenic R-groups Lemma 3.1.7 Let R be a right near-ring with identity which satisfies the DCCL. For any ideal P of R, P ? J1/2(R) = (0) if, and only if, P ? Soi(R). Proof From Lemma 2.3.3, P ? Soi(R) implies P ? J1/2(R) = (0). Conversely, suppose P ? J1/2(R) = (0). We show that P has an F - decomposition. The maximality of Soi(R) will then imply that P ? Soi(R). Putting J1/2(R) = J we have, by Lemma 3.1.1, R/J = (L1/J)? ? ? ? ? (Ln/J), CHAPTER 3. SOME CHARACTERIZATIONS 45 Li left ideals of R and Li/J of type-0 for i = 1, . . . , n. Reordering, if necessary, we have, by Lemma 2.2.2, R/J = ( (P ? J)/J ) ? (Lm+1/J)? ? ? ? ? (Ln/J). Since P is an ideal, PLj ? P ? Lj ? P ? J = (0), j = m + 1, . . . , n. The decomposition of the coset 1 + J gives 1+J = (u+J)+(xm+1+J)+? ? ?+(xn+J), u ? P, xj ? Lj, j = m+1, . . . , n. For any a ? P we have, by left distribution modulo J, a+ J = (au+ axm+1 + ? ? ?+ axn) + J. Thus a ? au ? J ? P = (0), because axj = 0, j = m + 1, . . . , n. Hence u is a right identity of P . We can also decompose the coset 1 + J as 1 + J = y1 + ? ? ?+ yn + J, yi ? Li, i = 1, . . . , n. Thus for any a ? P we have a+ J = (ay1 + ? ? ?+ aym + aym+1 + ? ? ?+ ayn) + J. Since ayj = 0, j = m + 1, . . . , n it follows that ?a + (ay1 + ? ? ? aym) ? J ? P = (0). Hence P = P ? L1 ? P ? L2 ? ? ? ? ? P ? Lm. Now if P?Li negationslash= 0, then J+(P?Li) is an R-kernel of Li properly containing J . Since Li/J is of type-0, this implies that Li = J + (P ? Li). In this CHAPTER 3. SOME CHARACTERIZATIONS 46 case we have Li/J = (J + P ? Li)/J ?= (P ? Li)/(P ? Li ? J) ?= P ? Li. Thus if P ? Li negationslash= 0, then it is of type-0. The decomposition of the right identity u of P namely, u = u1 + ? ? ?+ um, ui ? P ? Li, i = 1, . . . ,m gives an orthogonal idempotent set {ui} m i=1 because left distribution over the P ? Li is valid. Thus P has an F-decomposition. square Let ? be a faithful R-group and let S be the collection of all monogenic R-subgroups of ?. We define subclasses B(?), G(?) and K(?) of S as follows. Definition 3.1.8 Let R be a right near-ring with identity, and ? be a faithful R-group. Let S be the collection of all monogenic R-subgroups of ?. Define (a) K(?) to be the subclass of S of all R-groups of type-K, (b) B(?) to be the subclass of S of all type-0 R-groups which are R- isomorphic to subfactors of at least one member of K(?), (c) G(?) to be the subclass of S of all type-0 R-groups which are not in B(?). We note that the zero R-subgroup, (0), of ? is of type-0 as it has no non- trivial R-kernels, hence it is not of type-K. Thus the class K(?) is distinct from each of the classes, B(?) and G(?), of type-0 R-subgroups of ?. CHAPTER 3. SOME CHARACTERIZATIONS 47 When there is no ambiguity about the faithful R-group ?, we will simply write B, G and K, for the classes B(?), G(?) and K(?), respectively. Lemma 3.1.9 [3] Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and let R? be a monogenic R-subgroup of ?. If R? negationslash? K(?), then R? = ( ?m i=1 ?i) ? ?, where each summand is an R- kernel of R?, and each ?i is of type-0 (i = 1, 2, . . . ,m) and ? ? K(?). Definition 3.1.10 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL, and ? a faithful R-group. We define the ideal C(?) and the left ideal N(?) as follows C(?) = ? K?K(?) (0 : K), N(?) = ? ??? R??G(?) (0 : ?). Note. If K = ?, then C(?) = R. Since (0 : ?) is a maximal left ideal, it is clear that J1/2(R) ? N(?). Theorem 3.1.11 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL, and ? a faithful R-group. Then C(?) ? N(?) = (0) and C(?) ? Soi(R). Proof As J1/2(R) ? N(?), it follows that, if C(?) ? N(?) = (0), then C(?) ? J1/2(R) = (0), and the result will follow from Lemma 3.1.7. We now show that C(?) ? N(?) = (0). Let x ? C(?) ? N(?) and ? ? ?. If R? ? K, then x? = 0, by the definition of C(?). Otherwise, CHAPTER 3. SOME CHARACTERIZATIONS 48 R? = ( ?m i=1 ?i) ? ?, m ? 1, ?i of type-0, ? ? K by Lemma 3.1.9 and x? = (0). If ?i ? B, then x?i = 0, again by the definition of C(?). If ?i ? G, then ?i = R?i, and since ?i is of type-0, any R-subgroup R(r??i) is R?i or (0), for each r? ? R. Thus, if R(r??i) = R?i, then (0 : r??i) = (0 : ?i), so xr??i = 0. Otherwise, R(r??i) = (0) so again xr??i = 0. Hence, if x ? (0 : ?i), then x ? (0 : r??i), and x?i = (0). Now, let ?i be the component, in ?i, of the decomposition of ? as a sum ? = ?m j=1 ?j + ?, where ?j ? ?j, ? ? ?, j = 1, . . . ,m. Then x?i = 0 as x?i = (0), for all i = 1, 2, ...,m, and x? = 0. Thus x? = 0 and since ? is faithful it follows that x = 0. square Theorem 3.1.12 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL, and ? a faithful R-group. C(?) is a direct sum of elements from G. More precisely, C(?) is a direct sum of R-groups of type-0 each of which has an isomorphic copy in G. Proof Since R satisfies the DCCL, N := N(?) is an intersection of finitely many maximal left ideals. It follows from Lemma 3.1.1 that R/N = ? R?i?G R?i. But ( C(?) ? N ) /N is an R-kernel of R/N , and hence it follows readily from Lemma 2.2.2 that it is a direct sum of elements from G. The result now follows from the fact that C(?) ? N = (0), Theorem 3.1.11, and the CHAPTER 3. SOME CHARACTERIZATIONS 49 isomorphisms ( C(?)?N ) /N ?= C(?)/ ( C(?) ?N ) ?= C(?). square Theorem 3.1.13 Let R be a zero-symmetric right near-ring, with identity 1, satisfying the DCCL, and ? a faithful R-group. Then C(?) = Soi(R). Proof Since C(?) ? Soi(R) by Theorem 3.1.11, we need only show that Soi(R)K = (0), for every K ? K. Let Rk = K be in K and let R = Soi(R)? L be the socle-decomposition of R (cf. Definition 2.3.5.) Then K = Rk = Soi(R)k + Lk. (3.1) If Soi(R)k negationslash? Lk, then Reik negationslash? Lk for some type-0 summand Rei in the F-decomposition, Soi(R) = n? i=1 Rei. But then the map Rei ? Reik given by rei mapsto? reik is an R-isomorphism of Rei onto Reik, and thus Reik is of type-0 as Rei is of type-0. Since Reik ? Lk is an R-kernel of Reik, the fact that Reik is of type-0 and Reik negationslash? Lk, gives Reik ? Lk = (0). Thus, we can form a near-ring direct sum, Reik ? Lk. On the other hand, if Reik ? Lk, then Reik = Rei1eik ? ReiReik ? (ReiL)k = (0), hence Reik = (0). CHAPTER 3. SOME CHARACTERIZATIONS 50 Now, suppose Reik negationslash? Lk for i = 1, 2, . . . ,m, and Reik ? Lk for i = m+ 1, . . . , n, then Reik ? Lk = (0) and K ? Reik ? Lk if i = 1, 2, . . . ,m, and Reik = (0), i = m+1, . . . , n. If l ? {2, . . . ,m} and Relk ? Re1k?Lk, then discard Relk. Note that m ? 1, since Soi(R)k negationslash? Lk. So, after relabelling e2, . . . , em, without loss of generality, Rejk negationslash? Re1 ? Lk, for j ? {2, . . . ,m1}, where m1 ? m, and Rejk ? Re1 ?Lk, j = m1 + 1, . . . , n. Since Re2k is of type-0 and Re2k ? (Re1k ? Lk) is an R-kernel of Re2k, it follows that Re2k ? (Re1k?Lk) = (0), hence Re2k?Re1k?Lk ? K. Continuing this way we obtain K = t? j=1 Rejk ? Lk, m ? t ? 1. But this contradicts the fact that K has no R-group of type-0 as a near-ring direct summand. Thus Soi(R)K ? Lk and by (3.1), K ? Lk, so that Soi(R)K ? ( Soi(R)L ) k. But Soi(R)L ? Soi(R) ? L = (0), hence Soi(R)K = (0). Therefore Soi(R) ? C(?). square We remark that the above result shows that C(?) depends only on the near-ring R and not on the faithful R-group ?. Indeed, C(?) is precisely the Laxton-Machin critical ideal in the case of a d.g. near-ring R, cf. [12]. Also, we know that C(?) = Soi(R) is a direct sum of elements from G, but it is not clear whether each element from G has an isomorphic copy as a summand in Soi(R). Theorem 3.1.14 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL, and ? a faithful R-group. Let G and B be the classes of R-groups of type-0 as previously constructed in Definition 3.1.8, and R = Soi(R) ? L be the socle-decomposition of R. Each element of G has CHAPTER 3. SOME CHARACTERIZATIONS 51 an isomorphic copy as a summand of Soi(R) = C(?) and each element of B is a subfactor of L. Proof Let L be written as a near-ring direct sum L = ( m? i=1 Li ) ? ( r? j=1 L?j ) ? L??, where Li ? G, i = 1, . . . ,m, L?j ? B, j = 1, . . . , r, with m, r ? 0, and L?? ? K. Put M = Soi(R) ? ( ?m i=1 Li). Then MR = ( Soi(R)? m? i=1 Li )( Soi(R)? ( m? i=1 Li ) ? ( r? j=1 L?j ) ? L?? ) . (3.2) If m, r ? 1 and LiL?j negationslash= (0), then there exists x ? L ? j such that Lix negationslash= (0). It follows that the mapping Li ? Lix given by y mapsto? yx, for y ? Li, gives an R-isomorphism of Li onto Lix. Since Lix is an R-kernel of L?j, which is of type-0, and Lix negationslash= (0), we have Lix = L?j. But L ? j lessmuch K for some K ? K, consequently Li lessmuch K and thus Li ? B. This contradicts the fact that Li ? G and shows that LiL?j = (0), for all i = 1, . . . ,m, and j = 1, . . . , r. Now suppose m ? 1 and LiL?? negationslash= (0). Then there exists K ? K such that LiL??K negationslash= (0) as otherwise LiL?? ? Soi(R) giving LiL?? ? Soi(R) ? L?? = (0). Let k ? K and x ? L?? be such that Lixk negationslash= (0). Then the mapping Li ? Lixk, defined by li mapsto? lixk, for li ? Li, is an R- isomorphism of Li onto Lixk. Thus, Lixk is a type-0 R-kernel of K, but CHAPTER 3. SOME CHARACTERIZATIONS 52 then Lixk ? ( K/(Lixk) ) ?= K gives K a near-ring direct summand of type-0, a contradiction. Hence LiL?? = (0). Now consider (3.2) above. By right distribution and left distribution over a direct sum of R-kernels we have MR ? M so that M is a two-sided ideal. But M has an F-decomposition so that M ? Soi(R) by the maximality of Soi(R), i.e., m = 0. By Theorem 2.3.4, J1/2(R) ? L, and we may thus write the factor R-group R/J , J := J1/2(R), as R/J = (( Soi(R) + J ) /J ) ? ( L/J ) . Since every R-group ? of type-0 has an R-isomorphic copy as a direct sum- mand of R/J , Lemma 2.2.15 (ii), it follows that either ? is R-isomorphic to a summand of Soi(R) ?= ( Soi(R) + J ) /J or is a subfactor of L. By the above, since m = 0, if ? ? G, then ? is a summand of Soi(R). square Consider a type-0 R-subgroup, ? ? B(?), of a faithful R-group ?. There exists an R-isomorphism ? : ? ? K/H for some K ? K(?) and H an R-kernel of K. For any s ? Soi(R) and ? ? ? such that ?(?) = k +H, k ? K, we have ?(s?) = s?(?) = s(k +H) = sk +H = H. Since ? is an R-isomorphism, ?(0) = H, so that ?(0) = ?(s?), and hence s? = 0. This yields the following proposition. Proposition 3.1.15 Let R be a zero-symmetric right near-ring with iden- tity satisfying the DCCL, and ? a faithful R-group. For any ? ? B(?), Soi(R) ? (0 : ?). CHAPTER 3. SOME CHARACTERIZATIONS 53 3.2 A characterization of the nil-rigid series In this section, alternating chains, called ?-chains, of type-0 R-subgroups and type-K R-subgroups of a faithful R-group ? are considered. In Sub- section 3.2.2, an ascending series of ideals of R is defined in terms of the ?-chains. This series of ideals is called the ?-ideal series. In Subsection 3.2.3, it is proved that the ?-ideal series contains the nil-rigid series. 3.2.1 The class K(?) and the nil-rigid length The following results indicate a connection between the class, K(?) and ?R, the nil-rigid length of R. Proposition 3.2.1 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group. If ? is of type-s, then ?R = 1. Proof Since ? is faithful, J0(R) = Js(R) = (0). By Corollary 2.3.8, Soi(R) = R. Hence the nil-rigid series of R is : L1 = (0) subsetnoteql C1 = Soi(R) = R. So that ?R = 1. square CHAPTER 3. SOME CHARACTERIZATIONS 54 Proposition 3.2.2 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group. If ? is of type-0 but not of type-s, then ?R > 1. Proof As ? is faithful, J0(R) = (0). Since ? is of type-0 but not of type- s, it has a non-zero R-subgroup ? of type-K, by Lemma 3.1.4. Now, by Theorem 3.1.13, Soi(R) = C(?), thus the first two members of the nil-rigid series of R are such that L1 = J0(R) = (0) subsetnoteql C1 = Soi(R) ? (0 : ?) negationslash= R. Hence the nil-rigid series cannot end at C1. Consequently ?R > 1. square Proposition 3.2.3 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group. If K(?) = ?, then ?R = 1. Proof If K(?) = ?, then Soi(R) = R. This implies, by Corollary 2.3.8, that Js(R) = (0). Hence J0(R) = Js(R), and ?R = 1. square 3.2.2 The ?-chains In this subsection, ?-chains are defined and some of their characteristics are noted. The following lemma is an immediate consequence of Corollary 2.2.5. CHAPTER 3. SOME CHARACTERIZATIONS 55 Lemma 3.2.4 Let R be a zero-symmetric right near-ring with identity, and let ?1 supersetnoteql ?2 supersetnoteql ?3 supersetnoteql . . . supersetnoteql ?? (3.3) be a strictly descending chain of R-groups for some positive integer ?. Then ?i+1 is an (R/Ii)-group, where Ii = (0 : ?i), i = 1, 2, . . . , ? ? 1. Moreover, I1 ? I2 ? I3 ? . . . ? I?, is an ascending chain of ideals of R. Definition 3.2.5 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and let ? a faithful R-group. Let ?1 supersetnoteql ?2 supersetnoteql . . . supersetnoteql ???1 supersetnoteql ??, (3.4) be a strictly descending chain of R-subgroups of ? such that ?1 is maximal amongst type-0 R-subgroups of ?, and ?? is an R-group of type-s. The chain (3.4) is an ?-chain if for every pair (?i,?i+1), 1 ? i < ?, of consecutive members of chain (3.4), we have that: if ?i is of type-0, then ?i+1 is of type-K and it is maximal amongst type-K R-subgroups of ?i; and if ?i is of type-K, then ?i+1 is of type-0 and it is maximal amongst type-0 R-subgroups of ?i. We define the ?-length of the ?-chain (3.4) to be ??12 . We collect some facts about ?-chains in the next Proposition. CHAPTER 3. SOME CHARACTERIZATIONS 56 Proposition 3.2.6 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? be a faithful R-group. (a) Every type-0 R-subgroup of ? which is maximal amongst type-0 R- subgroups of ? is a member of some ?-chain. (b) If a type-0 R-subgroup of ? is maximal amongst type-0 R-subgroups of ? and is of type-s, then it forms a trivial ?-chain on its own. (c) If no type-0 R-subgroup of ? contains an R-subgroup of type-K, then the ?-length of every ?-chain is 0. (d) The ?-chains are alternating sequences of type-0 R-subgroups of ? and type-K R-subgroups of ?, starting with a type-0. (e) If a type-K R-subgroup, ?, of ?, is not contained in any type-0 R- subgroup of ?, then ? is not a member of any ?-chain. (f) If two type-K R-subgroups of ?, say ? and ??, are members of an ?-chain where ? supersetnoteql ??, then there exists a type-0 R-group, say ???, with ? supersetnoteql ??? supersetnoteql ??. We note, from Proposition 3.2.6 (d), that ? in (3.4) is odd. Examples of ?-chains and their lengths are given in Section 3.3. Theorem 3.2.7 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, ? a faithful R-group, and let ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ??, be an ?-chain, and L1 subsetnoteql C1 subsetnoteql L2 subsetnoteql C2 subsetnoteql . . . subsetnoteql L?R subsetnoteql C?R = R, CHAPTER 3. SOME CHARACTERIZATIONS 57 be the nil-rigid series of R. Then, the nil-rigid length, ?R, is such that ?R ? ? + 1 2 , and for any m = 1, 2, . . . , (? ? 1)/2, (a) ?2m is an (R/Lm)-group of type-K and (0 : ?2m) ? Cm, (b) ?2m+1 is an (R/Cm)-group of type-0 and (0 : ?2m+1) ? Lm+1, where Lm, Cm and Lm+1 are ideals of R in the nil-rigid series of R. Proof The proof follows by induction on m. For m = 1, note that (0 : ?1) ? J0(R) = L1, because ?1 is an R-group of type-0. Since ?1 supersetnoteql ?2, we have (0 : ?2) ? (0 : ?1), and hence (0 : ?2) ? L1. It follows by Proposition 2.2.4 that ?2 is an (R/L1)-group, and since ?2 is of type- K as an R-group, it is an (R/L1)-group of type-K, by Proposition 3.1.6. Since ?2 is an (R/L1)-group of type-K and Soi(R/L1) is the intersection of annihilators of all (R/L1)-groups of type-K, see Theorem 3.1.13, we have (0 : ?2)R/L1 ? Soi(R/L1). By Proposition 2.2.6 and the definition of the nil-rigid series, (0 : ?2)/L1 ? Soi(R/L1) = C1/L1, thus (0 : ?2) ? C1, giving R negationslash= C1 and ?R ? 2. Now consider ?3. Since ?3 subsetnoteql ?2, then (0 : ?3) ? (0 : ?2), consequently (0 : ?3) ? C1 and ?3 is an (R/C1)-group by Proposition 2.2.4. In addi- tion, since ?3 is of type-0 as an R-group, it is a type-0 (R/C1)-group by Proposition 2.2.4. As the 0-radical J0(R/C1) is the intersection of annihila- tors of all (R/C1)-groups, (0 : ?3)R/C1 ? J0(R/C1). By the definition of the CHAPTER 3. SOME CHARACTERIZATIONS 58 nil-rigid series, J0(R/C1) = L2/C1, as the nil-rigid length of R is ?R ? 2. By Proposition 2.2.6, (0 : ?3)/C1 ? J0(R/C1) = L2/C1, and hence (0 : ?3) ? L2. Assume the result holds for m ? 1, where 1 < m ? (? ? 1)/2. Then, ?R ? m, the R-group ?2m?2 is an (R/Lm?1)-group of type-K and (0 : ?2m?2) ? Cm?1, also the R-group ?2m?1 is an (R/Cm?1)-group of type-0 and (0 : ?2m?1) ? Lm. For the R-group ?2m, since ?2m subsetnoteql ?2m?1, then (0 : ?2m) ? (0 : ?2m?1). It follows from the fact that (0 : ?2m) ? (0 : ?2m?1) ? Lm and Proposition 2.2.4 that ?2m is an (R/Lm)-group. But ?2m is an R-group of type-K, and hence an (R/Lm)-group of type-K, by Proposition 3.1.6. Thus (0 : ?2m)R/Lm ? Soi(R/Lm) as the socle ideal, Soi(R/Lm), is the intersection of all (R/Lm)-groups of type-K. By Proposition 2.2.6 and the definition of the nil-rigid series, (0 : ?2m)/Lm ? Soi(R/Lm) = Cm/Lm, thus (0 : ?2m) ? Cm, and R negationslash= (0 : ?2m), Cm negationslash= R, giving ?R ? m+ 1. Now, since ?2m+1 subsetnoteql ?2m, we have (0 : ?2m+1) ? (0 : ?2m). Again, (0 : ?2m+1) ? (0 : ?2m) ? Cm and Proposition 2.2.4 implies that ?2m+1 is an (R/Cm)-group. Since ?2m+1 is an R-group of type-0, it follows from Proposition 2.2.4 that ?2m+1 is of type-0 as an (R/Cm)-group. But then CHAPTER 3. SOME CHARACTERIZATIONS 59 (0 : ?2m+1)R/Cm ? J0(R/Cm) by definition of the 0-radical, J0(R/Cm). By the definition of the nil-rigid series, J0(R/Cm) = Lm+1/Cm. Then by Proposition 2.2.6, (0 : ?2m+1)/Cm ? J0(R/Cm) = Lm+1/Cm. Hence (0 : ?2m+1) ? Lm+1. square From Theorem 3.2.7, we observe that for each faithful R-group ?, the lengths of the ?-chains have an upperbound of 2?R? 1. Hence we can find a maximum for the lengths of the ?-chains. Definition 3.2.8 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and let ? a faithful R-group. An ?-chain with the greatest ?-length is called a KG-chain of ?, and the ?-length of a KG- chain is called the KG-length of ?. Note. If ? is the KG-length of ? and ?R is the nil-rigid length of R, then by Theorem 3.2.7, ?R ? ? + 1. Since the KG-length is 0 if, and only if, J0(R) = Js(R), the next propo- sition follows from Lemma 2.3.13. Proposition 3.2.9 Let R be a zero symmetric right near-ring with identity satisfying the DCCS, ? a faithful R-group, and let ?R be the nil-rigid length of R. The KG-length of ?, ? = 0 if, and only if, ?R = 1. CHAPTER 3. SOME CHARACTERIZATIONS 60 Theorem 3.2.10 Let R be a zero symmetric right near-ring with identity satisfying the DCCS, and let ? be a faithful R-group. If the nil-rigid length of R is ?R = 2, then ? has a KG-length ? = 1. Proof Suppose ?R = 2, then by Theorem 3.2.7, ?R = 2 ? ?+1. Assume that ? = 0. Then every R-group of type-0 has no R-subgroup of type-K. It follows from Lemma 3.1.4 that every R-subgroup of ? which is maximal amongst type-0 R-subgroups of ? is of type-s. That is, J0(R) = Js(R), and by Lemma 2.3.13, ?R = 1, contradicting the supposition that ?R = 2. Therefore, ? = 1. square 3.2.3 ?-ideal series and the nil-rigid series In this subsection, the ?-ideal series is defined, and it is proved that the ?-ideal series contains the nil-rigid series in the following sense. Given two series of ideals I1 ? I2 ? ? ? ? ? Il?1 ? Il(I), (3.5) and J1 ? J2 ? ? ? ? ? Jl?1 ? Jl(J), (3.6) with their lengths such that l(I) ? l(J), we say the series (3.5) contains the series (3.6) if the corresponding ideals are such that Ii ? Ji, for each i = 1, 2, . . . , l(J). CHAPTER 3. SOME CHARACTERIZATIONS 61 Definition 3.2.11 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group. Let ??,1 supersetnoteql ??,2 supersetnoteql . . . supersetnoteql ??,(?(?)?1) supersetnoteql ??,?(?), ? ? ?, (3.7) be all ?-chains. Define the j-th ?-ideal, Tj, of R to be the intersection of annihilators of the j-th members of each ?-chain of length ?(?) ? j, i.e. Tj := ? ??? ?(?)?j (0 : ??, j) , j = 1, 2, . . . , ?(?), where ?(?) = max{ ?(?) |? ? ?}. We also define T0 := (0) and T?(?)+1 := R. Note. In the above definition ?(?) is always odd (as the ?-chains are alternating chains starting and ending with type-0 R-groups), and ?(?)?1 2 is the KG-length of ?. That is ?(?) is the length of the longest ?-chains. Proposition 3.2.12 Let R be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, and let ? be a faithful R-group. The ?-ideals form an ascending series of ideals of R. That is, T1 ? T2 ? T3 ? . . . ? T?(?). We refer to the series T1 ? T2 ? T3 ? . . . ? T?(?), of the j-th ?-ideals, j = 1, 2, . . . , ?(?), as the ?-ideal series. CHAPTER 3. SOME CHARACTERIZATIONS 62 Proposition 3.2.13 Let R be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, ? a faithful R-group, and let T1 the first ?-ideal of R. Then, J0(R) = T1 = L1, where L1 is as in the definition of the nil-rigid series. Proof Let ??,1, ? ? ?, be as defined in (3.7), then T1 = ? ??? (0 : ??,1). Since each ??,1 is of type-0, then T1 ? J0(R). Each ??,1 is a type-0 R-subgroup of ? which is maximal amongst type-0 R-subgroups of ?, and by Proposition 3.2.6(a), every such maximal type-0 R-subgroup of ? is a member of some ?-chain. Thus T1 annihilates all type-0 R-subgroups of ?, because every type-0 R-subgroup of ? is either maximal amongst type-0 R-subgroups of ? or it is contained in a type-0 R-group which is maximal amongst type-0 R-subgroups of ?. As a result T1 = J0(R). square Proposition 3.2.14 Let R be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, ? a faithful R-group, with KG-length ? ? 1, and let T2 be the second ?-ideal of R. Then T2/L1 ? Soi (R/L1) and T2 ? C1, where L1 and C1 are as in the definition of the nil-rigid series of R. Proof Let ??,2, ? ? ?, be as defined in (3.7), then by definition of ?-ideals, T2 = ? ??? ?(?)?2 (0 : ??,2), where each ??,2, ?(?) ? 2, is a type-K R-subgroup of ?. From Propositions 3.2.12 and 3.2.13, T2 ? T1 = L1. CHAPTER 3. SOME CHARACTERIZATIONS 63 Also, by Theorem 3.2.7, each ??,2 is an (R/L1)-group of type-K. But, for each faithful (R/L1)-group, the socle ideal, Soi(R/L1), is the intersection of the annihilators of all its (R/L1)-subgroups of type-K, see Theorem 3.1.13, hence T2/L1 ? Soi(R/L1). By the definition of the nil-rigid series, T2/L1 ? Soi(R/L1) = C1/L1, thus T2 ? C1. square Proposition 3.2.15 Let R be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, ? a faithful R-group with KG-length ? ? 1, and let T3 the third ?-ideal of R. Then T3/C1 ? J0 (R/C1) and T3 ? L2, where C1 and L2 are as in the definition of the nil-rigid series of R. Proof Let ??,3, ? ? ?, be as defined in (3.7), then by definition of the ?- ideal series, T3 = ? ??? ?(?)?3 (0 : ??,3), and T3 ? T2 ? C1 by Propositions 3.2.12 and 3.2.13. Since each ??,3, ?(?) ? 3, is a type-0 R-group, it follows from Theorem 3.2.7 that each ??,3 is an (R/C1)-group of type-0. By definition of the 0-radical, we have T3/C1 ? J0 (R/C1) . Since L2/C1 = J0(R/C1), then T3 ? L2. square Theorem 3.2.16 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group with KG-length ? ? 2 . Let T1 ? T2 ? T3 ? . . . ? T?(?), CHAPTER 3. SOME CHARACTERIZATIONS 64 be the ?-ideal series. For m = 1, 2, . . . , ?(?)?12 , (a) T2m/Lm ? Soi (R/Lm) and T2m ? Cm, (b) T2m+1/Cm ? J0 (R/Cm) and T2m+1 ? Lm+1, where Lm, Cm and Lm+1 are ideals of R in the nil-rigid series of R. Proof For m = 1, 2 the statement follows from Propositions 3.2.14 and 3.2.15, respectively. We proceed by induction on m. Assume that the statement holds for m, with m < ?(?)?12 . That is, T2m/Lm ? Soi (R/Lm) and T2m ? Cm. Also, T2m+1/Cm+1 ? J0 (R/Cm+1) and T2m+1 ? Lm+1. Now for m+ 1. Let ??,1 supersetnoteql ??,2 supersetnoteql . . . supersetnoteql ??,(?(?)?1) supersetnoteql ??,?(?), ? ? ?, be all ?-chains. By definition of the ?-ideal series, T2m+2 = ? ??? ?(?)?2m+2 (0 : ??, 2m+2). By Proposition 3.2.12 and the induction hypothesis, T2m+2 ? T2m+1 ? Lm+1. Since each ??,2m+2, ?(?) ? 2m + 2, is an R-group of type-K, it follows from Theorem 3.2.7 that each ??,2m+2 is an (R/Lm+1)-group of type-K. But for each faithful (R/Lm+1)-group, the socle ideal, Soi(R/Lm+1), is the CHAPTER 3. SOME CHARACTERIZATIONS 65 intersection of the annihilators of all its (R/Lm+1)-subgroups of type-K, see Theorem 3.1.13, hence T2m+2/Lm+1 ? Soi(R/Lm+1). By definition of the nil-rigid series, T2m+2/Lm+1 ? Soi(R/Lm+1) = Cm+1/Lm+1, thus T2m+2 ? Cm+1. Again, by definition of the ?-ideal series, T2m+3 = ? ??? ?(?)?2m+3 (0 : ??, 2m+3). By Proposition 3.2.12, we have T2m+3 ? T2m+2 ? Cm+1. Since each ??,2m+3, ?(?) ? 2m + 3, is an R-group of type-0, then by Theorem 3.2.7, each ??,2m+3 is an (R/Cm+1)-group of type-0. But the 0-radical, J0(R/Cm+1), is the intersection of annihilators of all (R/Cm+1)- groups of type-0, hence T2m+3/Cm+1 ? J0(R/Cm+1). By definition of the nil-rigid series, T2m+3/Cm+1 ? J0(R/Cm+1) = Lm+2/Cm+1, hence T2m+3 ? Lm+2. square The ?-ideal series and the nil-rigid series are therefore related as follows. CHAPTER 3. SOME CHARACTERIZATIONS 66 Corollary 3.2.17 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group. The series of ?-ideals, T1 ? T2 ? T3 ? . . . ? T?(?) ? T?(?)+1 = R, and the nil-rigid series of R, L1 subsetnoteql C1 subsetnoteql L2 subsetnoteql C2 subsetnoteql . . . subsetnoteql L?R subsetnoteql C?R = R, are related by T2m?1 ? Lm and T2m ? Cm, for 1 ? m ? ?(?)?12 . The fact that the ?-ideal series and the nil-rigid series are distinct is due to the existence of the quotient R-groups of the next proposition. This is illustrated in the Section 3.3. Proposition 3.2.18 Let R be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, and let ? be a faithful R-group of KG-length ? = ?(?)?12 . For any two consecutive members ?2m and ?2m+1, of a KG- chain, ?1 supersetnoteql ?2 supersetnoteql ?3 supersetnoteql . . . supersetnoteql ?2?+1, we have that, if ?2m+1 is an R-kernel of ?2m, then ?2m/?2m+1 is an (R/Cm)-group. Proof Since ?2m+1 is an R-kernel of ?2m, the quotient group, ?2m/?2m+1, is an R-group. By Theorem 3.2.7, (0 : ?2m) ? Cm. Hence ?2m is an CHAPTER 3. SOME CHARACTERIZATIONS 67 (R/Cm)-group by Proposition 2.2.4. Since (0 : ?2m/?2m+1) ? (0 : ?2m), we have (0 : ?2m/?2m+1) ? (0 : ?2m) ? Cm, thus ?2m/?2m+1 is an (R/Cm)-group by Proposition 2.2.4. square Lemma 3.2.19 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and let (0) ? L1 subsetnoteql C1 subsetnoteql L2 subsetnoteql C2 subsetnoteql ? ? ? subsetnoteql L?R subsetnoteql C?R = R, be the nil-rigid series of R. Then the nil-rigid length of R/Ci is ?R ? i, where 1 ? i < ?R . Proof By definition of the nil-rigid series, the quotient near-ring R/Ci has the nil-rigid series (0) ? L1(i) subsetnoteql C1(i) subsetnoteql L2(i) subsetnoteql C2(i) subsetnoteql ? ? ? subsetnoteql L?R (i) subsetnoteql C?R?i(i), where L1(i)/Ci is an ideal of R/Ci such that L1(i)/Ci = J0(R/Ci) = Li+1/Ci, so that L1(i) = Li+1. Also, C1(i)/L1(i) = Soi(R/L1(i)) = Soi(R/Li+1) = Ci+1/Li+1, hence C1(i) = Ci+1. Similarly, L2(i)/C1(i) = J0(R/C1(i)) = J0(R/Ci+1) = Li+2/Ci+1, CHAPTER 3. SOME CHARACTERIZATIONS 68 which means L2(i) = Li+2, and C2(i)/L2(i) = Soi(R/L2(i)) = Soi(R/Li+2) = Ci+2/Li+2, so that C2(i) = Ci+2. Therefore, L?R?i(i) = L?R and C?R?i(i) = C?R = R, because L?R?i(i)/C?R?(i+1)(i) = J0(R/C?R?(i+1)(i)) = J0(R/C?R?1) = L?R/C?R?1 and C?R?i(i)/L?R?i(i) = Soi(R/L?R?i(i)) = Soi(R/Li+(?R?i)) = C?R/L?R , and C?R/L?R = R/L?R . We conclude that the nil-rigid series of R/Ci is the nil-rigid series of R starting with Li+1 and ending with C?R = R, giving ?R/Ci ? ?R ? i. But since Ci ? C?R?1 subsetnoteql C?R = R, we get C?R?1/Ci negationslash= C?R/Ci = R/Ci. Hence, the nil-rigid length of R/Ci, ?R/Ci = ?R ? i. square Conjecture 3.1 Let R be a zero-symmetric right near-ring with identity satisfying the DCCS, and ? a faithful R-group. If the KG-length of ? is ?, then the nil-rigid length of R, ?R = ? + 1. CHAPTER 3. SOME CHARACTERIZATIONS 69 3.3 Examples on ?-chains In this section, examples of near-rings and ?-chains are given, these examples also illustrate the fact that the nil-rigid series and the ?-ideal series may be distinct. Here, the symbol ?G will denote the group theoretic direct sum. Example 3.3.1 Let ? = Z32; H1 = {0, 2, 4, . . . , 30}; H2 = {0, 4, 8, . . . , 28}; H3 = {0, 8, 16, 24}; H4 = {0, 16}. Define the near-ring R as follows, R := { f ? M0(?) | f(Hi) ? Hi, i = 1, 2, 3, 4; h? h? ? H2j ? f(h) ? f(h?) ? H2j, ?h, h? ? H2j?1, j = 1, 2 }. Then R is a near-ring with a multiplicative identity under point-wise addi- tion and map composition. We make the following observations. (a) The group ? = Z32 is a type-0 faithful R-group as it has no non- trivial R-kernels. (b) The subgroup H1 is a monogenic R-group with only one type-0 R- kernel, H2. Since H2 is not a direct summand of H1, then H1 is of type-K. (c) The subgroup H4 is a type-0 R-kernel to the monogenic R-group H3 and it is not a direct summand of H3, hence H3 is of type-K. There is only one ?-chain, namely ? supersetnoteql H1 supersetnoteql H2 supersetnoteql H3 supersetnoteql H4, which is a KG-chain of KG-length ? = 2, i.e. ?(?) = 5. The ?-ideal series is now, (0) = T0 = T1 ? T2 ? T3 ? T4 ? T5 ? T6 = R, CHAPTER 3. SOME CHARACTERIZATIONS 70 where T1 = (0 : ?) = (0), T2 = (0 : H1), T3 = (0 : H2), T4 = (0 : H3), and T5 = (0 : H4). The nil-rigid series of R is (0) = L1 subsetnoteql C1 subsetnoteql L2 subsetnoteql C2 subsetnoteql L3 subsetnoteql C3 = R, where L1 = (0 : ?) = (0), C1/L1 = Soi(R/L1) = (0 : H1), L2/C1 = J0(R/C1) = (0 : H2) ? (0 : H1/H2), C2/L2 = Soi(R/L2) = (0 : H3), L3/C2 = J0(R/C2) = (0 : H4) ? (0 : H3/H4) and C3/L3 = Soi(R/L3) = R/L3. Therefore C3 = R and ?R = 3 = ? + 1. However, the nil-rigid series and the ?-ideal series are distinct because T3 supersetnoteql L2 and T5 supersetnoteql L3. The next example shows that there exist type-K R-groups which are not members of any ?-chains; that is, the socle ideal Soi(R) is not a member of the nil-rigid series (Soi(R) negationslash= C1). Again, we have that the ?-ideal series is distinct from the nil-rigid series. Example 3.3.2 Let ? = Z8 ?G Z4, ?1 = {0, 4} ?G Z4, ?2 = {0, 2, 4, 6} ?G {0}, ?3 = {0} ?G Z4, ?4 = {0, 4} ?G {0}, CHAPTER 3. SOME CHARACTERIZATIONS 71 ?5 = {0} ?G {0, 2}. Define the near-ring R as follows, R := { f ? M0(?) | f(?j) ? ?j, j = 1, . . . , 5; ? ? ?? ? ?2 ? f(?)? f(??) ? ?2, ??, ?? ? ?; ? ? ?? ? ?4 ? f(?)? f(??) ? ?4, ? ?, ?? ? ?2; ?1 ? ?2 ? ?5 ? f(?1)? f(?2) ? ?5, ? ?1, ?2 ? ?3 }. Then R is a near-ring with a multiplicative identity under point-wise addi- tion and mapping composition. We make the following observations. (a) The group ? is a faithful monogenic R-group but it is not of type-0 because it has ?2 as an R-kernel. (b) The subgroup ?1 is a type-0 R-group which is not of type-s as it con- tains a monogenic R-group ?3 which in turn has a type-0 R-subgroup ?5, but ?5 is not a direct summand of ?3. Thus ?3 is of type-K. (c) The subgroup ?2 is a monogenic R-group containing a type-0 R- subgroup ?4, which is not a direct summand of ?2. Therefore ?2 is of type-K. (d) The near-ring has two other R-groups of type-2, ?3/?5 and ?2/?4. The R-group ? has one non-trivial ?-chain, ?1 supersetnoteql ?3 supersetnoteql ?5, and one trivial ?-chain consisting just of ?4. Hence ?2 is not a member of any ?-chain. Also, the KG-length is ? = 1. The ?-ideal series is {0} = T0 ? T1 ? T2 ? T3 ? T4 = R, CHAPTER 3. SOME CHARACTERIZATIONS 72 where T1 = (0 : ?4) ? (0 : ?1), T2 = (0 : ?3), and T3 = (0 : ?5). Note that, since (0 : ?3/?5) ? (0 : ?3), the quotient R-group ?3/?5 is an (R/C1)-group. The nil-rigid series of R is {0} ? L1 subsetnoteql C1 subsetnoteql L2 subsetnoteql C2 = R, where L1 = (0 : ?4) ? (0 : ?1) ? (0 : ?2/?4) = J0(R), C1/L1 = Soi(R/L1) = (0 : ?3), L2/C1 = J0(R/C1) = (0 : ?5) ? (0 : ?3/?5). Hence the nil-rigid length ?R = 2 = ?+1,. The two series are distinct as L1 negationslash= T1 and L2 negationslash= T3. Chapter 4 Annihilator Ideals in Mn(R) Ideals which are intersections of annihilators of monogenic R-groups (or Mn(R)-groups) are herein referred to as annihilator ideals. In Section 4.1, known theory on the connection between R-groups and Mn(R)- groups is collected. Also, two ways in which Mn(R) acts on ?n, where ? is a faithful R-group, are defined. It is well known that for some non-monogenic R-groups, direct sums of n copies of themselves are monogenic as Mn(R)- groups. Three classes of non-monogenic R-groups are defined according to the type of their corresponding Mn(R)-groups. That is, a non-monogenic R- group is said to be of ?n-form if its corresponding monogenic Mn(R)-group is of type-? (? = 0, s,K). 73 CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 74 In Section 4.2, a theory is developed on how annihilator ideals of R and their corresponding ideals of Mn(R) are related. As an example of an annihilator ideal, the theory in Section 4.2 is applied to the socle ideal in Section 4.3. Independent of the theory of annihilator ideals in Section 4.2, it is proved in Section 4.3 that Soi(Mn(R)) ? (Soi(R))?. Examples which illustrate relationships between socle ideals of R and Mn(R) are given in Section 4.4. 4.1 Modules in matrix near-rings In order to relate annihilator ideals in R to their corresponding ideals in Mn(R), we recall known results on the relationships between R-groups and Mn(R)-groups. Some of these results, by Van der Walt in [26], use a gen- eralized notion of a monogenic R-group called a connected R-group. These results are given below in the terminology of Meldrum and Meyer, in [15], which has been used throughout this thesis. Definition 4.1.1 Let R be a zero-symmetric right near-ring with identity. Suppose ? is an R-group. Then ? is said to be locally monogenic if for any finite subset H of ? there is an ? ? ? such that H ? R?. Note that a monogenic R-group is locally monogenic. Van der Walt defined a natural action of Mn(R) on ?n as follows. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 75 Lemma 4.1.2 [26] Let R be a zero-symmetric right near-ring with iden- tity. For ?, a locally monogenic R-group, ?n is a locally monogenic Mn(R)- group under the action U??1, ?2, . . . , ?n? := (U?r1, r2, . . . , rn?)? for U ? Mn(R), ?i, ? ? ?, ri ? R, and ri? = ?i. Theorem 4.1.3 [26] Let R be a zero-symmetric right near-ring with iden- tity. A locally monogenic R-group, ?, has no non-trivial R-subgroups (resp. R-kernels) if, and only if, ?n has no non-trivial Mn(R)-subgroups (resp. Mn(R)-kernels). Theorem 4.1.4 [26] Let R be a zero-symmetric right near-ring with iden- tity, and let ? be a locally monogenic R-group. Then any Mn(R)-kernel (resp. Mn(R)-subgroup) of ?n is of the form Hn, where H is an R-kernel (resp. R-subgroup) of ?. It is a consequence of Theorem 4.1.3 that, any R-group, R?, is of type- ? (? = 0, s, 2) as an R-group if, and only if, (R?)n is of type-? as an Mn(R)-group. We extend this result to R-groups of type-K. Corollary 4.1.5 Let R be a zero-symmetric right near-ring with identity, and let ? be a monogenic R-group. The R-group ? has a non-trivial R- kernel if, and only if, ?n has a non-trivial Mn(R)-kernel. That is, ? is not of type-0 if, and only if, ?n is not of type-0. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 76 Lemma 4.1.6 Let R be a zero-symmetric right near-ring with identity, and let ? be a monogenic R-group. The R-group ? has no near-ring direct summands of type-0 if, and only if, ?n has no near-ring direct summands of type-0 as Mn(R)-subgroups. Proof ? Suppose ?n has a near-ring direct summand of type-0, say ?n = D ? B, where B is an Mn(R)-kernel of type-0 and D is some Mn(R)-kernel. By Theorem 4.1.4, D = Hn and B = Sn, where both H and S are R-kernels of ?. Since Hn?Sn ?=Mn(R) (H?S) n, it follows that ? = H?S. Since ? is monogenic, by Proposition 3.1.2 both H and S are monogenic, and consequently S is of type-0, by Theorem 4.1.3. Thus ? has a near-ring direct summand of type-0. ? Conversely, suppose ? has a near-ring direct summand of type-0, say ? = H ? P, where P is an R-kernel of type-0 and H is an R-kernel. Since ? is monogenic, it follows from Proposition 3.1.2 that H is monogenic. Hence ?n, Hn and P n are monogenic Mn(R)-groups, by Lemma 4.1.2. It is easy to show that both Hn and P n are Mn(R)-kernels of ?n. By Theorem 4.1.3, P n is a type-0 Mn(R)-kernel of ?n. Since ?n = (H ? P )n ?=Mn(R) H n ? P n, it follows that ?n has a type-0 Mn(R)-group as a direct summand. square CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 77 The next theorem follows immediately from Corollary 4.1.5 and Lemma 4.1.6. Theorem 4.1.7 Let R be a zero-symmetric right near-ring with identity, and let ? be a monogenic R-group. The R-group ? is of type-K if, and only if, ?n is of type-K as an Mn(R)-group. The next result is a generalization of Lemma 1.6 of [20], and the proof extends easily, using Lemma 4.1.2, to any group, (?/?)n, where ? is a monogenic R-group and ? an R-kernel of ?. The action of Mn(R) on (?/?)n is defined by U?r1? + ?, r2? + ?, . . . , rn? + ?? := ?s1? + ?, s2? + ?, . . . , sn? + ??, where U?r1, r2, . . . , rn? = ?s1, s2, . . . , sn?, for some si ? R, 1 ? i ? n and U ? Mn(R). This action is well-defined, see Appendix A. Lemma 4.1.8 Let R be a zero-symmetric right near-ring with identity, ? a monogenic R-group, and let ? be an R-kernel of ?. Then (?/?)n ?=Mn(R) ? n/?n. Theorem 4.1.9 Let R be a zero-symmetric right near-ring with identity, ? a faithful R-group, and let ? and ? be monogenic R-subgroups of ?. Then, ? is an R-homomorphic image of ? if, and only if, ?n is an Mn(R)-homomorphic image of ?n. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 78 Proof ? Suppose ? is a homomorphic image of ?. Then there exists an R-homomorphism ? : ? ? ? such that ?/ker(?) ?=R ?. Clearly, ( ?/ker(?) )n ?=Mn(R) ? n. But, by Lemma 4.1.8, ( ?/ker(?) )n ?=Mn(R) ? n/(ker(?))n. Thus ?n/(ker(?))n ?=Mn(R) ? n. ? Conversely, suppose ? : ?n ? ?n is an Mn(R)-homomorphism such that ?n ?=Mn(R) ? n/ker(?). By Theorem 4.1.4, ker(?) = Hn, for H some R-kernel of ?. Thus, ?n ?=Mn(R) ? n/Hn. Again, by Lemma 4.1.8, ?n/Hn ?=Mn(R) ( ?/H )n . It follows that ( ?/H )n ?=Mn(R) ? n and hence ?/H ?=R ?. square From Theorems 4.1.3 and 4.1.9, it follows, under the conditions thereof, that for a faithful R-group, ?, if Mn(R) has DCCS, then ? ? B(?) iff ?n ? B(?n). (4.1) Also, if ? ? G(?) then ? is of type-0, making ?n a type-0 Mn(R)-group and ?n negationslash? B(?n), by (4.1). Thus ?n ? G(?n). CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 79 Conversely, ?n ? G(?n) implies that ?n is an Mn(R)-group of type-0 and ?n negationslash? B(?n). Since ? is monogenic, it is of type-0, by Theorem 4.1.3. So that ? negationslash? B(?), by (4.1), hence ? ? G(?). To summarize we have the following theorem. Theorem 4.1.10 Let R be a zero-symmetric near-ring with identity and ? a faithful R-group. For any monogenic R-subgroup ? of ?, (a) ? ? B(?) if, and only if, ?n ? B(?n), (b) ? ? G(?) if, and only if, ?n ? G(?n). The action of the matrix near-ring Mn(R) is not limited to Mn(R)-groups of the form ?n, where ? is locally monogenic as in Lemma 4.1.2. There is another action of Mn(R) on ?n called Action 2, for which ? need not be locally monogenic. Meldrum and Meyer used Action 2 to show that type-0 Mn(R)-groups exist in several non-isomorphic ways, see [15]. This action is explained below. Suppose ? = m? ?=1 ?? is a group theoretic direct sum of monogenic R-subgroups ?? = R??, ? = 1, 2, ...,m. Then, as a group theoretic direct sum, ?n := n? i=1 ( m? ?=1 ?? ) and there is a group isomorphism ? : ?n ? m? ?=1 ( n? i=1 ?? ) = m? ?=1 ?n? CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 80 defined by ?(??r11?1, r12?2, . . . , r1m?m?, ? ? ? , ?rn1?1, rn2?2, . . . , rnm?m??) = ??r11?1, . . . , rn1?1?, ?r12?2, . . . , rn2?2?, ? ? ? , ?r1m?m, . . . , rnm?m??. This can be rewritten as ??r11, . . . , rn1??1, ?r12, . . . , rn2??2, ? ? ? , ?r1m, . . . , rnm??m ?. Each ?r1j, . . . , rnj?, 1 ? j ? m, is an element in Rn on which a matrix acts naturally. Action 2 is now defined as follows. Definition 4.1.11 (Action 2) [15] Let R be a zero-symmetric right near- ring with identity, U ?Mn(R) and ?? = ??1, ?2, . . . , ?n? be any element in ?n. Then U?? = U??1, ?2, . . . , ?n? := ? ?1U?(??1, ?2, . . . , ?n?), is a well-defined Mn(R) action on ?n. Proposition 4.1.12 Let R be a zero-symmetric right near-ring with iden- tity. Let ? = m? i=1 R?i be an R-group which is a group theoretic direct sum of monogenic R-groups. If n ? m, then ?n is a monogenic Mn(R)-group under Action 2. Proof Consider the following matrices, V(r1,r2,..,rn) = f r1 11 + f r2 12 + ? ? ? + f rn 1n and U(a1,a2,..,an) = f a1 11 + f a2 21 + ? ? ? + f an n1 , for some ri, ai ? R, 1 ? i ? n. Then CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 81 ??1U(a1,a2,..,an)V(r1,r2,..,rn)????1, 0, . . . , 0?, ?0, ?2, 0, . . . , 0?, ? ? ? ? ? ? , ?0, . . . , 0, ?m?, ?0, . . . , 0?, ? ? ? , ?0, . . . , 0?? = ??1U(a1,a2,..,an)V(r1,r2,..,rn)???1, 0, . . . , 0?, ?0, ?2, . . . , 0?, ? ? ? , ?0, .., 0, ?m, 0, .., 0?? = ??1U(a1,a2,..,an)??r1?1, 0, . . . , 0?, ?r2?2, 0, . . . , 0?, ? ? ? , ?rm?m, 0, . . . , 0?? = ??1??a1r1?1, a2r1?1, . . . , anr1?1?, ? ? ? , ?a1rm?m, a2rm?m, . . . , anrm?m?? = ??a1r1?1, a1r2?2, . . . , a1rm?m?, ? ? ? , ?anr1?1, anr2?2, . . . , anrm?m??. Thus, for an arbitrary element in ?n, ??r11?1, r12?2, . . . , r1m?m?, ? ? ? , ?rn1?1, rn2?2, . . . , rnm?m?? = ??1( f 111 V(r11,r12,..,r1m) + ? ? ?+ f 1 n1 V(rn1,rn2..,rnm))?(?), where ? := ???1, 0, . . . , 0?, ?0, ?2, 0, . . . , 0?, ? ? ? , ?0, . . . , 0, ?m?, ?0, . . . , 0?, ? ? ? , ?0, . . . , 0??. Hence ? is an Mn(R)-generator of ?n. square We can now classify non-monogenic R-groups such as ? in Proposition 4.1.12, according to the type-? (? = 0, s,K) of their corresponding mono- genic Mn(R)-group ?n. Definition 4.1.13 Let ? be a non-monogenic R-group which is a group- theoretic direct sum of monogenic R-subgroups of ?. Let n ? N, n ? 1. (a) If ?n is an Mn(R)-group of type-0 but not of type-s, then ? is said CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 82 to be an R-group of 0n-form. (b) If ?n is an Mn(R)-group of type-s but not of type-2, then ? is said to be an R-group of sn-form. (c) If ?n is an Mn(R)-group of type-K, then ? is said to be an R-group of Kn-form. 4.2 Intersections of annihilators Jacobson-type radicals and the socle ideals in R, and in Mn(R), are intersec- tions of annihilators of monogenic R-groups, and monogenic Mn(R)-groups, respectively. Consequently we are able to develop a general theory relating annihilator ideals of R and the corresponding ideals of Mn(R). Lemma 4.2.1 Let R be a zero-symmetric right near-ring with identity. For any r ? R and monogenic R-group, ?, r ? (0 : ?) if, and only if, f rij ? (0? : ? n). Proof Let r be any non-zero element in R and ?? = ??1, ?2, . . . , ?n? be any element in ?n where ? = R?. For each ?i = ri?, with ri some element of R, we have f rij ?? = f r ij ?r1?, r2?, . . . , rn?? = f r ij ?r1, r2, . . . , rn? ? = ?0, . . . , 0, rrj, 0, . . . , 0? ? = ?0, . . . , 0, r?j, 0, . . . , 0? CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 83 where ri? appears in the i-th place. Clearly, r ? (0 : ?) if, and only if, f rij ? (0? : ? n). square Theorem 4.2.2 Let R be a zero-symmetric right near-ring with identity, and ? a faithful R-group. Let {R?? |? ? ?} be a collection of monogenic R-subgroups of ?, then ( ? ??? (0 : R??) )+ ? ? ??? (0? : (R??) n). Proof Let Q := ? ??? (0 : R??) and U be any matrix in Q+. We prove the result by induction on the weight of U . Firstly, let U = faij, where a ? Q. Then, by Lemma 4.2.1, faij ? ? ??? (0? : (R??) n). Secondly, let U = fa1ij (f a2 j k + f a3 j l ), where a1, a2, a3 ? Q, and let ?? = ??1, ?2, . . . , ?n? ? (R??)n. We now have U?? = ?0, . . . , 0, a1(a2?k + a3?l), 0, . . . , 0?. Clearly, U?? = 0? because a2, a3 ? Q and ?? ? (R??)n. The above two cases provide a basic step to an inductive proof based on the fact that the ideal Q+ is generated by { faij ? ? ? ? ? a ? Q = ? ??? (0 : R??), 1 ? i, j ? n } . CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 84 Assume that the theorem holds for any matrix U in Q+ of weight less than n, where n is a positive integer. Suppose U = V + W where V and W are matrices in Q+ each of a weight less than n. Then, for any R??, ? ? ?, and ?? = ??1, ?2, . . . , ?n? ? (R??)n, we have U?? = V ?? + W?? = 0? + 0?. Thus, U ? Q+. Suppose U = VW where V and W are matrices in Q+ each of a weight less than n. Then, for any R??, ? ? ?, and ?? = ??1, ?2, . . . , ?n? ? (R??)n, we have U?? = V ( W (??) ) = V (0?) = 0?. Hence the result follows. square Let ? a faithful R-group, and let E?(R) denote the class of all type-? R-subgroups of ?, ? = 0, s, 2,K, and E?(Mn(R)) denote the class of all type-? Mn(R)-subgroups of ?n, ? = 0, s, 2,K. Theorem 4.2.3 Let R be a zero-symmetric right near-ring with identity, and ? a faithful locally monogenic R-group. Then ? ??E?(Mn(R)) (0? : ?) ? ? ??E?(R) (0? : ?n). Moreover, if ? has no R-subgroups of ?n-form, then the two ideals are equal. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 85 Proof By Theorems 4.1.3 and 4.1.7, ? ? E?(R) ? ? n ? E?(Mn(R)), giving E?(Mn(R)) ? { ?n | ? ? E?(R) }. Hence ? ??E?(Mn(R)) (0? : ?) ? ? ??E?(R) (0? : ?n), which proves the result. square If a faithful locally monogenic R-group ? of a zero-symmetric near-ring, R, with identity, has no R-subgroups of ?n-form, then every type-? Mn(R)- subgroup, ?, of ?n, is of the form ? = ?n, for some R-group ?, see Theorem 4.1.3. Hence E?(Mn(R)) = { ?n | ? ? E?(R) } and ? ??E?(Mn(R)) (0? : ?) = ? ??E?(R) (0? : ?n). This and Theorem 4.2.2 yield the next corollary. Corollary 4.2.4 Let R be a zero-symmetric right near-ring with identity, and ? a faithful locally monogenic R-group. If ? has no R-group of ?n- form, ? = 0, s,K, then ? ? ? ??E?(R) (0 : ?) ? ? + ? ? ??E?(Mn(R)) (0? : ?). CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 86 Lemma 4.2.5 Let R be a zero-symmetric right near-ring with identity, and ? an R-group. For each ? ? ? we have (0 : R?)? ? (0? : (R?)n). Proof Let ?? ? (R?)n, say ?? = ??? where ?? = ?r1, r2, r3, . . . , rn? ? Rn. For W ? Mn(R) we write W?? = ?r ? 1, r ? 2, r ? 3, . . . , r ? n? ? R n. Suppose W ? (0 : R?)?, then W?? = ?r ? 1, r ? 2, r ? 3, . . . , r ? n? ? (0 : R?) n. Equivalently, r?i ? (0 : R?) for each 1 ? i ? n. Now, in terms of ?? we have W?? = W ?r1?, r2?, r3?, . . . , rn?? = ?r ? 1, r ? 2, r ? 3, . . . , r ? n??, where each r?i ? (0 : R?), giving W?? = 0?. Since ?? ? (R?) n was arbitrary W ? (0 : (R?)n). square Corollary 4.2.6 Let R be a zero-symmetric right near-ring with identity. For any monogenic R-group, R?, such that |R?| = 2, (0 : R?)? = (0 : (R?)n). Proof By Lemma 4.2.5 we need only prove that (0 : (R?)n) ? (0 : R?)?. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 87 Let W ? (0 : (R?)n). Assume that W negationslash? (0 : R?)?. Then there exists some ?b1, b2, b3, . . . , bn? in Rn such that W ?b1, b2, b3, . . . , bn? negationslash? (0 : R?)n. That is, if W ?b1, b2, b3, . . . , bn? = ?b ? 1, b ? 2, b ? 3, . . . , b ? n? then there exists j ? {1, 2, .., n} such that b?j negationslash? (0 : R?), but since |R?| = 2, we have (0 : R?) = (0 : ?). This gives W ?b1, b2, b3, . . . , bn?? = ?b ? 1?, b ? 2?, b ? 3?, . . . , b ? n?? negationslash= 0?, which contradicts W ? (0 : (R?)n). square Note that in Lemma 4.2.5, if |R?| negationslash= 2 and R? is not faithful, then (0 : ?) may strictly contain (0 : R?), in which case, (0 : R?)? subsetnoteql (0 : (R?)n). Lemma 4.2.7 Let R be a zero-symmetric right near-ring with identity, and ? be an R-group. For each ? ? ?, (0? : (R?)n) ? (0 : ?)?. Proof Let U ? (0? : (R?)n) and ?? be an element of Rn, then ?? := ?? ? ? (R?)n. Hence 0? = U?? = U?? ?, giving U?? ? (0 : ?)n. Thus U ? (0 : ?)?. square We now extend the earlier results to R-groups which are direct sums of R-groups of order 2. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 88 Consider the R-group, ? = m? i=1 R?i, where each R?i is an R-kernel of ?, and |R?i| = 2. By left distributivity over R-kernels (cf. Lemma 2.3.1) (0 : ?) = m? i=1 (0 : R?i). Again, by Theorem 2.4.6 (iii), ( m? i=1 (0 : R?i) )? = m? i=1 (0 : R?i) ? . Hence (0 : ?)? = m? i=1 (0 : R?i) ?. (4.2) On the other hand, ?n = ( m? i=1 R?i )n ?=Mn(R) m? i=1 ( R?i )n , and by left distributivity over Mn(R)-kernels, (0? : ?n) = m? i=1 (0? : (R?i) n). By Corollary 4.2.6, (0? : (R?i) n) = (0 : R?i) ?, for each i = 1, 2, . . . ,m. Thus (0? : ?n) = m? i=1 (0 : R?i) ?. (4.3) The next theorem follows from (4.2) and (4.3). Theorem 4.2.8 Let R be a zero-symmetric right near-ring with identity, and let ? be an R-group such that ? = m? i=1 R?i. If each R-subgroup R?j is an R-kernel of ? and a group of order 2, then (0? : ?n) = (0 : ?)?. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 89 Theorem 4.2.9 Let R be a zero-symmetric right near-ring with identity, and let ? be an R-group such that ? = m? i=1 R?i. If each R-subgroup R?j is an R-kernel of ? and a group of order 2, then (0 : ?)? = ( m? i=1 (0 : ?i) )? . Proof Since (0? : ?n) = (0 : ?)?, by Theorem 4.2.8, we need only show that (0? : ?n) = ( m? i=1 (0 : ?i) )? . We note that (0 : R?i) = (0 : ?i) as |R?i| = 2, i = 1, 2, . . . ,m. Hence (0 : R?i) ? = (0 : ?i) ?. By Corollary 4.2.6, (0? : (R?i) n) = (0 : R?i) ?, for each i = 1, 2, . . . ,m. By left distributivity over Mn(R)-kernels, (0? : ?n) = m? i=1 (0? : (R?i) n), as ?n ?=Mn(R) m? i=1 ( R?i )n . Hence (0? : ?n) = m? i=1 (0 : ?i) ?, and the result follows by Proposition 2.4.6 (iii). square CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 90 4.3 The ideals Soi(R)+ and Soi(R)? By Theorem 3.1.13 the socle ideal is an annihilator ideal. The containment of Soi(Mn(R)) in (Soi(R))? is established here, and we apply the theory de- veloped in the previous section to study the relationship between (Soi(R))+ and Soi(Mn(R)). Since (Soi(R))+ = ( ? ??K(?) (0? : ?) )+ , the following theorem is a conse- quence of Theorem 4.2.2. Theorem 4.3.1 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCL, and let ? be a faithful locally mono- genic R-group. Then (Soi(R))+ ? ? ??K(?) (0 : ?n). According to Theorem 3.1.13, the socle ideal is characterizable as Soi(Mn(R)) = ? ??K(?n) (0? : ?). The next theorem is a consequence of Theorem 4.2.3. Theorem 4.3.2 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCL, and let ? be a faithful locally mono- genic R-group. Then Soi(Mn(R)) ? ? ??K(?) (0? : ?n). CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 91 The next theorem in a direct consequence of Theorem 4.1.7 and Theorem 4.3.2 with ? = K, along with Theorem 4.3.1. Theorem 4.3.3 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCL, and let ? be a faithful locally mono- genic R-group. If ? has no R-groups of K-form, then Soi(Mn(R)) = ? ??K(?) (0 : ?n). Moreover, (Soi(R))+ ? Soi(Mn(R)). In general, there is no relationship between (Soi(R))+ and Soi(Mn(R)). This is illustrated by Example 4.4.3, where an example is given in which (Soi(R))+ negationslash? Soi(Mn(R)). The next theorem is a consequence of Proposition 2.4.6 (iii) and Lemma 4.2.5. Theorem 4.3.4 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCL, and let ? be a faithful locally mono- genic R-group. Then (Soi(R))? ? ? ??K(?) (0 : ?n). We now apply the theory of F-decompositions of a near-ring, R, as in Section 3.1, to study the relationship between (Soi(R))? and Soi ( Mn(R)) for near-rings satisfying the DCCL. Recall that, R = Soi(R) ? L, if R satisfies the DCCL. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 92 Theorem 4.3.5 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL. Let R = Soi(R)? L be the socle-decomposition of R, and ? be a faithful locally monogenic R-group. If Mn(R) satisfies the DCCL, then (Soi(R))? = (0? : Ln), the annihilating ideal of Ln. Proof Since R = Soi(R) ? L, each r in R is of the form r = s + l where s ? Soi(R) and l ? L. Clearly Rn = (Soi(R))n ? Ln and any ?? := ?r1, r2, . . . , rn? in Rn is of the form ?? = ?s1, . . . , sn? + ?l1, . . . , ln? where si ? Soi(R) and li ? L, 1 ? i ? n. Since ( Soi(R) )n and Ln are Mn(R)-kernels of Rn, for any U ? Mn(R), U?? = U [?s1, . . . , sn?+ ?l1, . . . , ln?] = U?s1, . . . , sn?+ U?l1, . . . , ln?, where U?s1, . . . , sn? =: ?s?1, . . . , s ? n? ? (Soi(R)) n and U?l1, . . . , ln? =: ?l?1, . . . , l ? n? ? L n, also we may write U?? = ?s?1, . . . , s ? n?+ ?l ? 1, . . . , l ? n? =: ?s ?? 1, . . . , s ?? n?. If U ? (Soi(R))? = { V ? Mn(R) | V ?? ? ( Soi(R) )n , ? ?? ? Rn } , then U?? ? (Soi(R))n, giving l?i = s ?? i ? s ? i ? Soi(R) ? L = (0), ? 1 ? i ? n. Thus U?l1, . . . , ln? = ?l?1, . . . , l ? n? = ?0??. That is, (Soi(R)) ? ? (0? : Ln). CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 93 Conversely, if U ? (0 : Ln), then U?l1, . . . , ln? = ?0?? for any ?l1, . . . , ln? in Ln. This implies that U?? = U?s1, . . . , sn? ? (Soi(R))n for all ?? ? Rn. Therefore U ? (Soi(R))? so that (0 : Ln) ? (Soi(R))?. square Theorem 4.3.6 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL. Let R = Soi(R)? L be the socle-decomposition of R, and ? be a faithful locally monogenic R-group. If Mn(R) satisfies the DCCL, then Soi(Mn(R)) ? (Soi(R))?. Proof From Theorem 4.3.5, we need only show that Soi ( Mn(R) ) Ln = (0?). By Theorem 3.1.14, since R = Soi(R)?L, the left ideal L may be written as a direct sum L = ( r? j=1 L?j ) ? L??, where L?j ? B(?) and L ?? ? K(?). Now Rn = (Soi(R))n ? Ln and thus Ln = ( r? j=1 (L?j) n ) ? (L??)n. By Theorems 4.1.10 and 4.1.7, each (L?j) n ? B(?n) and (L??)n ? K(?n). Since Soi(Mn(R)) annihilates all Mn(R)-groups of type-K and all Mn(R)- groups in B(?n), from Proposition 3.1.15, it follows that Soi ( Mn(R) ) Ln = (0?). square The inclusion in Theorem 4.3.6 above can be strict as shown in the Example 4.4.3. That is, there are cases when Soi (Mn(R)) subsetnoteql (Soi(R))?. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 94 Corollary 4.3.7 Let R be a zero-symmetric right near-ring with identity satisfying the DCCL. Let R = Soi(R)? L be the socle-decomposition of R, and ? be a faithful locally monogenic R-group. If ? has no R-subgroups of K-form, then Soi(Mn(R)) = (Soi(R)) ? = ? ??K(?) (0 : ?n). Proof By Theorem 4.3.6, Soi(Mn(R)) ? (Soi(R))?, and by Theorem 4.3.4, (Soi(R))? ? ? ??K(?) (0 : ?n). But, Soi (Mn(R)) = ? ??K(?) (0? : ?n) by Theorem 4.3.3. Hence the result follows. square 4.4 Examples on Socle Ideals In this section, examples are given to illustrate the relationships between (Soi(R))+ and Soi(Mn(R)), and between (Soi(R))? and Soi(Mn(R)), in the context of the theory of Section 4.3. We give examples of near-rings where the above ideals are not comparable and examples for which they are comparable and the inclusion is strict. The symbol ?G is used to denote the group theoretic direct sum. The subgroup of a group, G, generated by x? ? G is denoted by grp{x?}. The following example illustrates that the inclusion in Theorem 4.3.1 can be strict. That is, (Soi(R))+ subsetnoteql ? ??K(?) (0? : ?n). CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 95 Example 4.4.1 Let ? := Z2 ?G Z2 ?G Z2, ?1 := {(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0)}, ?2 := {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)}, ?3 := grp{(0, 1, 0)} and ?4 := grp{(0, 0, 1)}. Define a near-ring R by R := { f ? M0(?) | f(?i) ? ?i, i = 1, 2, 3, 4; ? ?, ?? ? ?2, ? ? ?? ? ?j ? f(?)? f(??) ? ?j, where j = 3, 4; ? ?1, ?2 ? ?1, ?1 ? ?2 ? ?3 ? f(?1)? f(?2) ? ?3 }. Then R is a near-ring with a multiplicative identity under point-wise addi- tion and map composition. We observe the following facts about R: 1. The group ? is a faithful R-group of type-0. Thus, J0(R) = (0). 2. The subgroups ?1 and ?3 are monogenic R-groups. The R-subgroup ?3 is an R-kernel of both ?1 and ?2. The R-subgroup ?4 is an R-kernel of ?2. 3. The R-group ?1 is monogenic but has no R-group of type-0 as a direct summand, therefore it is of type-K. As a result the faithful R-group ? is not of type-s, and hence Js(R) negationslash= (0). 4. The classes of monogenic R-subgroups of ? are; K(?) = {?1}, B(?) = {?3,?1/?3}, G(?) = {(0),?,?2,?4}, (?4 ?= ?2/?3). 5. The socle ideal Soi(R) = (0 : ?1), and Js(R) = (0 : ?4) ? (0 : ?3) ? (0 : ?1/?3) = J2(R). Consider the matrix near-ring M2(R). We show that there exists a matrix U in (0? : ?21) which is not in (Soi(R)) +. Let U = f c11(f a 12 + f b 11) where CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 96 a, b, and c are maps in R defined by a b c (1, 1, 1) mapsto? (1, 0, 1), (1, 1, 1) mapsto? (1, 1, 1), (1, 1, 1) mapsto? (1, 1, 1), (1, 0, 1) mapsto? (1, 0, 1), (1, 0, 1) mapsto? (1, 1, 1), (1, 0, 1) mapsto? (1, 0, 1), (0, 1, 1) mapsto? (0, 0, 1), (0, 1, 1) mapsto? (0, 0, 1), (0, 1, 1) mapsto? (0, 1, 1), (0, 0, 1) mapsto? (0, 0, 1), (0, 0, 1) mapsto? 0?, (0, 0, 1) mapsto? (0, 0, 1), (0, 1, 0) mapsto? 0?, (0, 1, 0) mapsto? (0, 1, 0), (0, 1, 0) mapsto? 0?, (1, 0, 0) mapsto? (0, 1, 0), (1, 0, 0) mapsto? (0, 1, 0), (1, 0, 0) mapsto? (0, 1, 0), (1, 1, 0) mapsto? (0, 1, 0), (1, 1, 0) mapsto? (0, 1, 0), (1, 1, 0) mapsto? (0, 1, 0), 0? mapsto? 0?. 0? mapsto? 0?. 0? mapsto? 0?. We note that a, b, c negationslash? (0 : ?1), where ?1 := {0?, (0, 1, 0), (1, 0, 0), (1, 1, 0)}, so that faij, f b ij and f c ij are not in (0 : ?1) + = (Soi(R))+, i, j ? {1, 2}. Consider the elements r? and r of R, defined below, and note the resulting near-ring element c(ar?+ br), where a, b and c are as previously defined. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 97 r? r c(ar? + br) (1, 1, 1) mapsto? (1, 1, 1), (1, 1, 1) mapsto? (1, 0, 1), (1, 1, 1) mapsto? 0?, (1, 0, 1) mapsto? (1, 0, 1), (1, 0, 1) mapsto? (1, 1, 1), (1, 0, 1) mapsto? 0?, (0, 1, 1) mapsto? (0, 0, 1), (0, 1, 1) mapsto? (0, 1, 1), (0, 1, 1) mapsto? 0?, (0, 0, 1) mapsto? (0, 0, 1), (0, 0, 1) mapsto? (0, 0, 1), (0, 0, 1) mapsto? (0, 0, 1), (0, 1, 0) mapsto? 0?, (0, 1, 0) mapsto? (0, 1, 0), (0, 1, 0) mapsto? 0?, (1, 0, 0) mapsto? (1, 0, 0), (1, 0, 0) mapsto? (0, 1, 0), (1, 0, 0) mapsto? (0, 1, 0), (1, 1, 0) mapsto? (1, 1, 0), (1, 1, 0) mapsto? 0?, (1, 1, 0) mapsto? (0, 1, 0), 0? mapsto? 0?. 0? mapsto? 0?. 0? mapsto? 0?. Then ?r, r?? ? R2 and we have U?r, r?? = ?c(ar? + br), 0?. Observe that c(ar? + br) negationslash? (0 : ?1) = Soi(R). Thus U negationslash? (Soi(R))?. Since (Soi(R))+ ? (Soi(R))?, we conclude that U negationslash? (Soi(R))+. The above example also illustrates that (Soi(R))+ can be strictly contained in Soi(Mn(R)), as noted below. Example 4.4.2 The near-ring R given in Example 4.4.1 has no R-groups of K2-form, therefore K(?2) = {?21} and Soi(M2(R)) = (0 : ? 2 1). Thus, (Soi(R))+ is strictly contained in Soi(M2(R)). The next example gives a near-ring, R, for which (Soi(R))+ negationslash? Soi (Mn(R)) . Indeed, the inequality or lack of containment between the two ideals is due to the presence of R-groups of K2-form. This example further illustrates that Soi(Mn(R)) may be strictly contained in (Soi(R))?. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 98 Example 4.4.3 Let ? := Z2 ?G Z4 ?G Z2; ?1 := grp{(1, 0, 0), (0, 2, 0), (0, 0, 1)}; ?2 := grp{(0, 1, 0)}; ?3 := grp{(0, 2, 0), (0, 0, 1)}; ?4 := grp{(1, 0, 0), (0, 2, 0)}; ?1 := grp{(1, 0, 0)}; ?2 := grp{(1, 2, 0)}; ?3 := grp{(0, 2, 0)}; ?4 := grp{(0, 0, 1)}; ?5 := grp{(0, 2, 1)}. Define a near-ring R as follows, R := { f ? M0(?) | f(?i) ? ?i, i = 1, 2, 3, 4; f(?j) ? ?j, j = 1, 2, .., 5; ??, ?? ? ?l, (l = 2, 3)? ? ?? ? ?3 ? f(?)? f(??) ? ?3 }. Then R is a near-ring with a multiplicative identity under point-wise addi- tion and map composition. Each of the subgroups ?i (i = 1, 2, 3, 4) and ?j (j = 1, 2, 3, 4, 5), is an R-subgroup of ?. All these R-groups are monogenic, except for ?3 and ?4. In addition, ?3 is an R-kernel of both ?2 and ?3. Hence, ?3 is of K2-form, and ?4 is of 02-form. The classes of monogenic R-subgroups of R with respect to ? are K(?) = {?2}, B(?) = {?3}, G(?) = { (0),?,?1,?j (j = 1, 2, 4, 5) }. The socle ideal of R is therefore Soi(R) = (0 : ?2). We now consider ?2 as a monogenic faithful M2(R)-group (because ? is monogenic as an R-group). We also note that ?23 and ? 2 4 are monogenic M2(R)-groups. CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 99 The classes of monogenic M2(R)-subgroups of ?2 are K(?2) = {?22,? 2 3}, B(? 2) = {?23,? 2 4,? 2 5}, G(?2) = {(0?), ?2, ?21, ? 2 1, ? 2 2, ? 2 4}. Thus, the socle ideal is Soi (M2(R)) = (0? : ?22) ? (0? : ? 2 3) . Consider the maps w and u in R defined by w u (0, 3, 0) mapsto? (0, 3, 0), (0, 3, 0) mapsto? (0, 2, 0), (0, 1, 0) mapsto? (0, 1, 0), (0, 1, 0) mapsto? (0, 2, 0), (0, 2, 0) mapsto? 0?, (0, 2, 0) mapsto? 0?, (0, 0, 1) mapsto? (0, 0, 1), (0, 0, 1) mapsto? (0, 0, 1), (0, 2, 1) mapsto? (0, 2, 1), (0, 2, 1) mapsto? 0?, (1, 0, 0) mapsto? 0?, (1, 0, 0) mapsto? (1, 0, 0), (1, 2, 0) mapsto? (1, 2, 0), (1, 2, 0) mapsto? (1, 2, 0), 0? mapsto? 0?. 0? mapsto? 0?, x? mapsto? x?, ? x? negationslash? (?2 ? ?3 ? ?4), x? mapsto? x?, ? x? negationslash? (?2 ? ?3 ? ?4). We note that (wu)y? = (w ? u)y? = 0?, for all y? ? ?2. Meanwhile, wu(0, 0, 1) = (0, 0, 1) negationslash= 0?. Thus, wu negationslash? (0 : ?3). Hence, the matrix U := fwu12 is an element of (Soi(R))+ but not of Soi (M2(R)) . CHAPTER 4. ANNIHILATOR IDEALS IN MN (R) 100 Let us also note that the matrix U = fwu12 ? (Soi(R)) ?, but not in Soi (M2(R)). square Chapter 5 ?-Primitivity in Mn(R) In this chapter, the connection between ?-primitive ideals (? = 0, s, 2) of R and Mn(R) is considered. A similar connection is investigated for the s-socle ideal, which is the minimal ideal modulo which the Jacobson s-radical is nilpotent. 5.1 ?-Radicals of Mn(R) Here we apply the theory of annihilator ideals of Section 4.2 to the ?- primitive ideals. Let E?(R) be the class of all type-? R-subgroups of a faithful locally monogenic R-group, and E?(Mn(R)) be the class of type-? Mn(R)-subgroups of a faithful locally monogenic Mn(R)-group, (? = 0, s, 2). 101 CHAPTER 5. ?-PRIMITIVITY IN MN(R) 102 Denote by P(?), the ideal ? R? ?E?(R) (0? : (R?)n). Collecting the results from Section 4.2, we obtain that the ideal P(?) relates to ?-primitive ideals of Mn(R) as follows. Theorem 5.1.1 Let R be a zero symmetric right near-ring with identity. Then, for ? = 0, s, 2, (a) (J?(R))? ? P(?), (b) J?(Mn(R)) ? P(?), (c) (J?(R))+ ? P(?). It is well known that if ? = 2 the inclusions in parts (a) and (b) of the above theorem are in fact equality, see Theorem 2.4.11 (i). Since I+ ? I?, for any ideal I of R, it now follows that (J2(R)) + ? (J2(R)) ? = J2(Mn(R)) = P(2). For ? = 0, s, the inclusions (a) and (b) may be strict. Also, J?(Mn(R)) ? (J?(R))?, for ? = 0, s, and thus J?(Mn(R)) ? (J?(R))? ? P(?), for ? = 0, s. (5.1) If a faithful locally monogenic R-group, ?, of R has no R-subgroups of ?n-form, then E?(Mn(R)) = { ?n | ? ? E?(R) } CHAPTER 5. ?-PRIMITIVITY IN MN(R) 103 for ? = 0, s, and hence, by Theorem 4.2.3, J?(Mn(R) = P(?). This together with (5.1) yield the next proposition. Proposition 5.1.2 Let R be a zero symmetric right near-ring with identity, and ? a faithful locally monogenic R-group. For ? = 0, s, if ? has no R-subgroups of ?n-form then J?(Mn(R)) = (J?(R))? = P(?). The ideal P(?) turns out to be helpful when constructing examples of near- rings with either a strict inclusion between two annihilator ideals of Mn(R), or no inclusion at all. Meldrum and Meyer conjectured that (J0(R))+ ? J0(Mn(R)) in [17]. However, counter examples to this conjecture exist, see Example 5.2.1. The next theorem, a direct consequence of Corollary 4.2.4, shows that there is, subject to some conditions, a relationship between (J0(R))+ and J0(Mn(R)). Theorem 5.1.3 Let R be a zero symmetric right near-ring with identity and ? a faithful locally monogenic R-group, and let ? = 0, s. If ? has no R-subgroups of ?n-form, then (J?(R)) + ? J?(Mn(R)). We will now proceed to prove that (Js(R))+ ? Js(Mn(R)) when Mn(R) has DCCL. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 104 Let R be a zero-symmetric near-ring with identity such that Mn(R) satisfies the DCCL. Then, by Theorems 2.2.3 and 4.1.4, any Mn(R)-group, ?, of type-s is a homomorphic image of some monogenic Mn(R)-subgroup of Rn, say Kn. Let ? : Kn ? ? be the Mn(R)-homomorphism and Ln := ker(?). Then Kn/Ln ?=Mn(R) ?. By Lemma 4.1.8, (K/L)n ?= Kn/Ln. (5.2) Lemma 5.1.4 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCL, and let Kn/Ln be an Mn(R)-group of type-s. Then (0 : K/L) is an intersection of s-primitive ideals of R. Proof. For k ? K, let M := (Mn(R))(?k, 0, ..., 0?+Ln), a monogenic Mn(R)-subgroup of Kn/Ln. Now Kn/Ln is an Mn(R)-group of type-s, hence M = l? i=1 Mi, where each Mi is of type-0 and is an Mn(R)-kernel of M . Then ?k, 0, ..., 0?+ Ln = (?k11, ..., kn1?+ L n) + ...+ (?k1l, ..., knl?+ L n), where each ?k1i, ..., kni?+Ln is an Mn(R)-generator of Mi. Applying the matrix f 111 to the above gives ?k, 0, ..., 0?+ Ln = (?k11, 0, ..., 0?+ L n) + ...+ (?k1l, 0, ..., 0?+ L n), where ?k1j, 0, . . . , 0? + Ln ? Mj, which in fact generates Mj. It follows readily that (Rk + L)/L = (Rk11 + L)/L? ? ? ? ? (Rk1l + L)/L, CHAPTER 5. ?-PRIMITIVITY IN MN(R) 105 where, by Theorem 4.1.3, each (Rk1j +L)/L is of type-0 and an R-kernel of (Rk + L)/L. Thus (0 : k + L) = l? j=1 (0 : k1j + L) is the intersection of maximal left ideals. Consequently, (0 : K/L) = ? k?K (0 : k + L), an intersection of s-primitive ideals, by Theorem 2.2.12. square Theorem 5.1.5 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies DCCL. Then (Js(R)) + ? Js(Mn(R)). Proof. Let ? be an Mn(R)-group of type-s. Then ? ?=Mn(R) K n/Ln, for some Mn(R)-subgroups Kn, and Ln of Rn. By Lemma 5.1.4, (0 : K/L) ? Js(R). Thus, for each f?ij ? (Js(R)) +, ? ? Js(R), and ?k1, .., kn? ? Kn, we have f?ij(?k1, .., kn?+ L n) = ?0, .., 0, ?kj, 0, .., 0?+ L n = ?0, 0, . . . , 0?+ Ln. Hence f?ij ? (0 : K n/Ln), and therefore f?ij ? (0 : ?). Since ? is arbitrary, we have f?ij ? Js(Mn(R)). square The inclusion in Theorem 5.1.5 may be strict as shown in Example 1 of [8]. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 106 5.2 Examples on ?-radicals In this section we give an example of a near-ring R where 1. J0(Mn(R)) negationslash= Js(Mn(R)) negationslash= J2(Mn(R)) while J0(R) = Js(R) = J2(R), and 2. (J0(R))+ negationslash? J0(Mn(R)) and Js(Mn(R)) subsetnoteql (Js(R))?. The symbol ?G is used to denote a group theoretic direct sum. Example 5.2.1 Consider the group ? := Z2 ?G Z4 ?G Z2 ?G Z2, and the following subgroups: S := Z2 ?G Z4 ?G {0} ?G {0}; ? := {0}?G {0, 2}?GZ2?G {0}; H1 := {0?, (1, 0, 0, 0), (0, 2, 0, 0), (1, 2, 0, 0)}; H2 := {0?, (0, 1, 0, 0), (0, 2, 0, 0), (0, 3, 0, 0)}; T1 := {0?, (1, 0, 0, 0)}; H3 := {0?, (0, 2, 0, 0), (1, 1, 0, 0), (1, 3, 0, 0)}; T2 := {0?, (0, 2, 0, 0)}; T3 := {0?, (1, 2, 0, 0)}; T4 := {0?, (0, 0, 1, 0)}; T5 := {0?, (0, 2, 1, 0)}. Define a subnear-ring R of M0(?) as R := { f ? M0(?) | f(S) ? S; f(?) ? ?; f(Hi) ? Hi, ? i = 1, 2, 3; f(Tj) ? Tj, ?j = 1, 2, .., 5; ? ? ?? ? ? ? f(?)? f(??) ? ?,??, ?? ? ?; h? h? ? T2 ? f(h)? f(h?) ? T2,?h, h? ? Hl, (l = 1, 2, 3) }. Then R is a near-ring with a multiplicative identity under point-wise addi- tion and mapping composition. Observe that: 1. ? = R(1, 1, 1, 1) is not of type-0 as ? is a non-trivial R-kernel of ?. Therefore J0(R) negationslash= (0). 2. H2 and H3 are R-groups of type-K. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 107 3. S = T1?G H2 is an R-group of 02-form. That is, S2 is an M2(R)-group of type-0 but not of type-s. 4. ? = T2?G T4 is of s2-form. That is, ?2 is an M2(R)-group of type-s but not of type-2. 5. The type-0 R-subgroups, Tj (j = 1, 2, .., 5), of ? are of type-2. The quotient R-groups, Hl/T2 (l = 1, 2, 3), are of type-0 and of order 2 as groups, thus of type-2. Since every type-0 R-group has an isomorphic copy in ?, these are all the R-groups of type-0, up to isomorphism. Therefore J0(R) = Js(R) = J2(R). (a) We show that (J0(R))+ negationslash? J0(M2(R)). Define a map z in R by z(x) = ? ???? ???? 0 if x ? (H1 ? ?) (0, 2, 0, 0) if x ? (Hl \ T2), l = 2, 3 x if x ? ? \ (S ? ?) . Note that z annihilates all type-0 R-groups, thus z ? J0(R). Hence, for each i, j, f zij ? (J0(R)) +. For s1 = (1, 3, 0, 0) ? S, f z12?s1, s1? = ?zs1, 0? = ?(0, 2, 0, 0), 0? negationslash= ?0, 0?. Thus f z12 /? (0 : S 2). Since S2 is a type-0 M2(R)-group, it follows that f z12 /? J0(M2(R)). (b) Here we show that J0(M2(R)) negationslash= Js(M2(R)). Consider the element z of R as defined above, and let V := f z11 + f z 12. For any ?r1, r2? ? R2, V ?r1, r2? = ?zr1 + zr2, 0?. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 108 For ?s1, 0? ? S2, where s1 = (1, 3, 0, 0), we have V ?s1, 0? = ?zs1, 0? = ?(0, 2, 0, 0), 0? negationslash= ?0, 0?. So V negationslash? J0(M2(R)). The type-s M2(R)-groups are ?2, T 2l , (Hl/T2) 2 ?=M2(R) H 2 l /T 2 2 , l = 1, 2, 3. By definition V ? ((0 : ?) ? (0 : H1)) + , and hence V ? (0 : ?)+ ? (0 : H1) + ? (0 : ?2) ? (0 : H21 ) by Proposition 2.4.6, and Lemma 4.2.1, respectively. A simple calculation shows that V (H2l ) ? T 2 2 and hence V (H2l /T 2 2 ) ? 0? + T 2 2 for each l = 1, 2, 3. Therefore V ? Js(M2(R)). (c) We now prove that Js(M2(R)) negationslash= J2(M2(R)). Define an R-subgroup of (R,+) as K := {0, k1, k2, k3} where ki(x) = ? ? ? bi if x = (0, 0, 0, 1) 0 if x negationslash= (0, 0, 0, 1) and b1 = (0, 2, 0, 0), b2 = (0, 0, 1, 0) and b3 = (0, 2, 1, 0). CHAPTER 5. ?-PRIMITIVITY IN MN(R) 109 Note that K is of s2-form as it is R-isomorphic to ? = T2 ? T4. That is, K2 is an M2(R)-group of type-s. Define elements t and s of R by t(b3) = b3 and t(x) = 0 otherwise, and s(b1) = b1, s(b2) = b2 and s(x) = 0, otherwise. The matrix B := f t11(f s 11 + f s 12) on R 2 yields B?r1, r2? = ?t(sr1 + sr2), 0?. For ?r1, r2? = ?k1, k2?, observe that B?k1, k2? = ?t(sk1 + sk2), 0? and t(sk1 + sk2)(0, 0, 0, 1) = t(sb1 + sb2) = t(b1 + b2) = t(b3) = b3 negationslash= 0. Thus, B negationslash? (0 : K2). That is, there exists an M2(R)-group of type-s which B does not annihilate. Therefore B negationslash? Js(M2(R)). Simple calculations show that, if ?b, b?? ? ( 5? i=1 T 2i ) , then B?b, b?? = ?0, 0?, and if ?b, b?? ? ( 3? l=1 H2l /T 2 2 ) , CHAPTER 5. ?-PRIMITIVITY IN MN(R) 110 then B?b, b?? ? T 22 . That is, B annihilates every M2(R)-group of type-2, therefore B ? J2(M2(R)). (d) In addition, since (Js(R)) ? = (J2(R)) ? = J2(M2(R)), and Js(M2(R)) negationslash= J2(M2(R)), we conclude that Js(M2(R)) subsetnoteql (Js(R))?. 5.3 The s-socles Recall that the s-socle of R is the minimal ideal, A, of R modulo which Js(R) is non-zero nilpotent, see Section 2.3. Note that, by definition the s-socle, A, is contained in Js(R), hence A+ ? (Js(R))+, by Proposition 2.4.6. For the remainder of this thesis the s-socle of R will be denoted by A, while the s-socle of Mn(R) will be denoted A. For a subset J of R we denote J (m) = { r1r2 ? ? ? rm | ri ? J, i = 1, 2, . . . ,m }. The next proposition is a direct consequence of Lemma 2.4.7 and the defini- tion of the s-socle of R. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 111 Proposition 5.3.1 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Then (Js(R/A))+ is nilpotent if, and only if, Js(R/A) is nilpotent. Theorem 5.3.2 [26, Lemma 4.2] Let R be a zero-symmetric right near- ring with identity satisfying the DCCS, and let I be an ideal of R. Then Mn(R/I) ?= Mn(R)/I?. Let I and J be non-trivial ideals of a zero-symmetric right near-ring with identity, R, such that J ? I. Let x, xij, z ? J , i, j = 1, 2, . . . , n, r ? R and consider the map ? : J+ ? (J/I)+ defined recursively by n? i,j=1 fxijij mapsto? n? i,j=1 f (xij + I)ij , and f rp? ( n? i,j=1 fxijij ) mapsto? f (r+ I)p? ( n? i,j=1 f (xij + I)ij ) . This map, ?, is the restriction to J+ of the map used in the proof of Lemma 4.2 in [26]. Thus ? is well-defined and is a near-ring homomorphism as shown in [26]. It follows easily that ? is surjective and that ker(?) = J+ ? I?, so that ker(?) ? I+. That ker(?) ? I+ is established in Appendix B. The following lemma is a direct consequence of the above. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 112 Lemma 5.3.3 Let R be a zero-symmetric right near-ring with identity, and let I and J be non-trivial ideals of R such that J ? I. Then J+/I+ ?= (J/I)+. Recall that Js(R/A) is non-zero and nilpotent, by Lemma 2.3.14. Also, Js(R/A) = Js(R)/A. It now follows from Proposition 5.3.1 that (Js(R/A))+ is nilpotent. By Lemma 5.3.3, (Js(R)/A)+ ?= (Js(R))+/A+, which yields the following result. Theorem 5.3.4 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Then (Js(R))+ is nilpotent modulo A+. By (i) and (iv) of Proposition 2.4.9, we have the following theorem. Proposition 5.3.5 [27] Let R be a zero-symmetric right near-ring with identity. If D is an ideal of Mn(R), then (D?) + ? D ? (D?) ?. This enables us to prove the following lemma. Lemma 5.3.6 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. If D is an ideal of Mn(R) contained in (Js(R))+ such that [(Js(R))+]/D is nilpotent, then A+ ? D. Proof For any ? ? Js(R), we have f?ij ? (Js(R)) +. Since [(Js(R)) +](m) ? D, CHAPTER 5. ?-PRIMITIVITY IN MN(R) 113 for some positive integer m, we have (f?ij) m ? D. For each ?r1, ..., rn? ? Rn, we have pii(f ? ii) m?r1, ..., rn? = pii?0, ..., ? mri, ..., 0? = ? mri ? D?. In particular, pii(f ? ii) m?1, ..., 1? = pii?0, ..., ? m, ..., 0? = ?m ? D?. Thus Js(R)/D? is a nil ideal of R/D?. Since R satisfies the DCCS, Js(R)/D? is nilpotent. By the minimality of A we have A ? D?. Therefore, by Propositions 2.4.6 (i) and 5.3.5, A+ ? (D?) + ? D. square Hence by Theorem 5.3.4 and Lemma 5.3.6 the next theorem follows. Theorem 5.3.7 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Then A+ is the unique minimal ideal modulo which (Js(R))+ is non-zero nilpotent. Since (Js(R))+ ? Js(Mn(R)), by Theorem 5.1.5, and by definition, Js(Mn(R))/A is nilpotent, it follows that (Js(R))+ + A is nilpotent modulo A. This implies that (Js(R))+ is nilpotent modulo A+ ? A. By the minimality of A+ we have the following theorem. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 114 Theorem 5.3.8 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Then A+ ? A. Clearly, A+ ? A?, but it is not clear what the relationship between A? and A is. However, if A = (0), then A? = (0). To show this we need the following lemma. Lemma 5.3.9 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS, and let ? be a monogenic R-group. If ? is an R-group of type-0 but not of type-s, then the Mn(R)-group ?n is also of type-0 but not of type-s. Proof Suppose ?n is an Mn(R)-group of type-s, and let (R?)n be any monogenic Mn(R)-subgroup of ?n. Then, by definition of type-s, Def- inition 2.2.1 (ii), along with Theorem 4.1.4, (R?)n is a direct-sum of the form (R?)n = k? i=1 Hni , where each Hni is an Mn(R)-kernel of type-0, for i = 1, 2, ....k. It follows from Lemma 4.1.6 that R? = k? i=1 Hi, where each Hi is an R-kernel of type-0. That is, ? is of type-s. square Theorem 5.3.10 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. If A = (0), then A? = (0). CHAPTER 5. ?-PRIMITIVITY IN MN(R) 115 Proof Since A = (0), it follows that Js(Mn(R)) = J0(Mn(R)), by definition of the s-socle and Lemma 2.3.13. Assume that A negationslash= (0). Then J0(R) negationslash= Js(R), by definition of the s-socle and Lemma 2.3.13, and hence there exists an R-group ? of type-0 which is not of type-s. It follows from Lemma 5.3.9 that ?n is an Mn(R)-group of type-0 which is not of type-s. This contradicts the fact that Js(Mn(R)) = J0(Mn(R)). Therefore J0(R) = Js(R), hence A = (0). It now follows that A? = (0). square Thus, if A? negationslash= (0), then A negationslash= (0). In fact, examples of near-rings exist where A properly contains A?, see Example 5.4.1. One would expect the connection between the s-socle ideals A and A to imitate that of their corresponding s-radicals, namely, A+ ? A ? A?. However, it turns out to be more complicated. Examples show that there is, in general, no relationship between A? and A. Only in restricted cases does one find a relationship between these two ideals, one such case is given below in Theorem 5.3.12. To this end we note, below, Corollary 2 to Theorem 4.4 of [6] as a lemma. Lemma 5.3.11 [6] Let R be a zero-symmetric right near-ring with identity satisfying the DCCS. If ?R = 2 and A ? Soi(R), then Js(R) ? Soi(R) = A. Theorem 5.3.12 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. If ?R = ?Mn(R) = 2, A ? Soi(R) and CHAPTER 5. ?-PRIMITIVITY IN MN(R) 116 A ? Soi(Mn(R)), then A ? A?. Proof By Lemma 5.3.11, A = Js(R) ? Soi(R) and A = Js(Mn(R)) ? Soi(Mn(R)). By Proposition 2.4.6 (iii), we have A? = [Js(R) ? Soi(R)] ? = (Js(R)) ? ? (Soi(R))?. Also, by Theorem 2.4.11 (ii), A ? Js(Mn(R)) ? (Js(R))?, and by Theorem 4.3.6, A ? Soi(Mn(R)) ? (Soi(R))?. Therefore, A ? A?. square 5.4 Examples on s-socles In this section, examples of near-rings are given to illustrate the relationship between the s-socle ideal of Mn(R) and the ideals A+ and A?, where A is the s-socle of R. The symbol ?G is used to denote a group theoretic direct sum. The following example illustrates that A? can be strictly contained in A. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 117 Example 5.4.1 Let G := Z2?GZ4?GZ2?GZ2, S ?? := Z2?GZ4?GZ2?G{0}, and S ? := {0} ?G {0} ?G Z2 ?G Z2. Let S1 := {(0, 0, 0, 0), (0, 2, 1, 1)}, S2 := {(0, 0, 0, 0), (0, 2, 0, 1)}, S3 := {(0, 0, 0, 0), (0, 2, 1, 0)}, S4 := {(0, 0, 0, 0), (0, 0, 1, 1)}, S5 := {(0, 0, 0, 0), (0, 0, 0, 1)}, S6 := {(0, 0, 0, 0), (0, 0, 1, 0)}, S7 := {(0, 0, 0, 0), (0, 2, 0, 0)}. Define a near-ring R by R := {f ? M0(G) | f(S ??) ? S ??, f(Si) ? Si,?i ? {1, 2, . . . , 7}; ?s, s? ? S ?, s? s? ? S5 ? f(s)? f(s?) ? S5; ?g, g? ? G, g ? g? ? S ?? ? f(g)? f(g?) ? S ?? }. We note that (a) Each Si is an R-subgroup of G, for each i ? {1, 2, . . . , 7}. (b) G is a monogenic R-group generated by (1, 3, 1, 1), and S ?? is an R- kernel of G. Thus G is not of type-0, so that J0(R) negationslash= (0). (c) S ?? is an R-group of type-s but not of type-2, because it has R- subgroups S3, S6 and S7 none of which is an R-kernel. Thus J0(R) negationslash= J2(R). (d) S ? = S6 ?G S5 is a non-monogenic R-subgroup of G because all its proper subgroups are R-groups. (e) S5 is an R-kernel of S ?. (f) All the type-0 R-subgroups of G are also of type-s, hence Js(R) = J0(R). Therefore, the s-socle A of R is zero. Hence, A? = (0). CHAPTER 5. ?-PRIMITIVITY IN MN(R) 118 We now look at the R-subgroups of (R,+), but first we define the following maps in R. Let x? := (1, 2, 1, 1), and 0? := (0, 0, 0, 0). Define k0 to be the zero-map; k1(x?) := (0, 2, 1, 1); k2(x?) := (0, 2, 0, 1); k3(x?) := (0, 2, 1, 0); k4(x?) := (0, 0, 1, 1); k5(x?) := (0, 0, 0, 1); k6(x?) := (0, 0, 1, 0); k7(x?) := (0, 2, 0, 0); such that kj(y?) = 0?, if y? negationslash= x?, for each j ? {1, 2, . . . , 7}. Define K := {k0, k1, k2, . . . , k7 }, H := {k0, k4, k5, k6}, and T := {k0, k5}, and observe that (1) K and H are non-monogenic R-groups, and can be written as group- theoretic direct-sums as follows ; K = Rk6 ?G Rk5 ?G Rk7, and H = Rk6 ?G Rk5. (2) The maps kj, for j = 1, 2, . . . , 7, are defined in such a way that we have the following R-isomorphisms between R-subgroups of (R,+) and the R-subgroups of G Rk6?=R S6; T = Rk5?=R S5; Rk7?=R S7 and S ??=R H. (3) T = Rk5 is an R-kernel of H. Consider K2 as an M2(R)-group under Action 2. Let ?k, k?? ? K2, where k = a1k6 + a2k5 + a3k7 and k ? = b1k6 + b2k5 + b3k7. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 119 We may rewrite k and k? as k = ?a1k6, a2k5, a3k7? and k ? = ?b1k6, b2k5, b3k7?. There is an isomorphism ? : K2 ? K2 defined by ?(?k, k??) = ??a1k4, b1k4?, ?a2k5, b2k5?, ?a3k7, b3k7??, which is an element of (Rk6)2 ?G (Rk5)2 ?G (Rk7)2. A matrix U ?M2(R) acts on K2 by Action 2 as follows U?k, k?? = ??1U??k, k?? = ??1?(U?a1, b1?)k4, U(?a2, b2?)k5, U(?a3, b3?)k7?. Taking U := fa11 + f b 21 for a, b ? R, we obtain U?k6 + k5 + k7, 0? = ? ?1?U?1, 0?k6, U?1, 0?k5, U?1, 0?k7? = ??1??ak6, bk6?, ?ak5, bk5?, ?ak7, bk7?? = ??ak6, ak5, ak7?, ?bk6, bk5, bk7??. Since a and b can be arbitrarily chosen in R, all the 64 elements of K2 can be obtained by using ?k6 + k5 + k7, 0? as an M2(R)-generator. So then K2 = M2(R)?k6 + k5 + k7, 0?. In a similar manner, it can be shown that the subgroup H2 of K2 is a monogenic M2(R)-group under Action 2, such that H2 = M2(R)?k6 + k5, 0?. We note that K2 has no M2(R)-kernels, hence it is an M2(R)-group of type-0. Since H2 is a monogenic M2(R)-subgroup of K2 which is not CHAPTER 5. ?-PRIMITIVITY IN MN(R) 120 a direct sum of M2(R)-kernels, it follows that K2 is not of type-s. In particular, H2 is of type-K as it has only one M2(R)-kernel T 2. Hence Js(M2(R)) negationslash= J0(M2(R)), therefore the s-socle A of M2(R) is non-zero. From this it follows that A? = (0) subsetnoteql A. The next example illustrates that, in general, the ideals A? and A are not comparable. For the near-ring in this example we observe that A? negationslash? A. Example 5.4.2 Let ? := Z4 ?G Z2 ?G Z4; ?1 := Z4 ?G Z2 ?G {0}; ?2 := {0} ?G Z2 ?G Z4; ?1 := {0?, (2, 0, 0), (0, 1, 0), (2, 1, 0)}; ?2 := {0?, (1, 0, 0), (2, 0, 0), (3, 0, 0)}; ?3 := {0?, (0, 1, 0), (0, 0, 2), (0, 1, 2)}; T1 := {0?, (0, 1, 0)}; T2 := {0?, (2, 0, 0)}; T3 := {0?, (2, 1, 0)}; T4 := {0?, (0, 0, 2)}; T5 := {0?, (0, 1, 2)}. Define a near-ring, R, by R := { f ? M0(?) | f(?i) ? ?i, i = 1, 2; f(?j) ? ?j, j = 1, 2, 3; f(Tl) ? Tl, l = 1, 2, . . . , 5; ? ?, ?? ? ?j (j = 1, 2), ? ? ?? ? T2 ? f(?)? f(??) ? T2 }. 1. Observations on R-subgroups of ?. (a) It follows immediately from definition of R that the group ? is a faithful R-group, and ? is of type-0 because it has no non-trivial R-kernels. But, CHAPTER 5. ?-PRIMITIVITY IN MN(R) 121 ? contains a monogenic R-group, ?2, which is of type-K as ?2 has no R-group of type-0 as a direct-summand and is itself not of type-0 because it has an R-kernel, T2. This implies ? is not of type-s, and hence, J0(R) = (0) negationslash= Js(R), and therefore the s-socle A negationslash= (0). (b) The subgroup ?1 is of type-0 because it has no R-kernels. However, ?1 contains ?2, therefore ?1 is not of type-s just as ? is not of type- s. The group ?2 is a type-s R-group as it has no R-kernels and all its monogenic R-subgroups are of type-2. (c) The monogenic R-group ?2 is in K(?), because it has no type-0 R-group as a direct-summand. The type-0 R-subgroups of ? are : ?2, Tj, j = 1, 2, 3, 4, 5, and all other R-groups of type-0 are isomorphic to ?i/T2, i = 1, 2. (d) The subgroups ?1 and ?3 are both non-monogenic R-subgroups of ?. The R-group T2 is an R-kernel of ?1, while ?3 has no R-kernels. In fact, ?1 is of K2-form, whereas ?3 is of s2-form. (e) The classes of monogenic R-subgroups of ? are: K(?) := {?2}; B(?) := {T2}; G(?) := {(0), ?, ?1, ?2, T1, T3, T4, T5 }. The Socle ideal of R is therefore given by Soi(R) = (0 : ?2). (f) The nil-rigid series of R is (0) = L1 ? C1 ? L2 ? C2 = R CHAPTER 5. ?-PRIMITIVITY IN MN(R) 122 where L1 = J0(R) = (0), C1 = Soi(R) = (0 : ?2), L2/C1 = J0 (R/C1) ? (0 : T2), and C2/L2 = Soi (R/L2) = R/L2, because K (R/L2) = ?. Therefore the nil-rigid length of R, is ?R = 2, and hence the s-socle A ? C1 = Soi(R). It follows from Lemma 5.3.11 that A = Js(R) ? Soi(R), and hence A = [ 5? j=1 (0 : Tj) ] ? [ 2? i=1 (0 : ?i/T2) ] ? (0 : ?2) ? (0 : ?2). Since ?1 = T1 ?G T2, in fact ?1 = T1 ? T2 ? T3, the s-socle is A = [ 5? j=1 (0 : Tj) ] ? (0 : ?2) ? (0 : ?2). 2. Classifications of M2(R)-subgroups of ?2. (a) Let us first note that ?2 is a faithful monogenic M2(R)-group. Also, ?22 is a monogenic M2(R)-group containing a type-0 M2(R)-group T 2 2 . But, T 22 is not a direct-summand of ? 2 2. Hence, ? 2 is not of type-s, it is however of type-0 as it has no M2(R)-kernels. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 123 (b) Each M2(R)-group H2, where H is a monogenic R-subgroup of ?, has the same properties that H had as an R-group. As for the non- monogenic R-subgroups ?1 and ?3, their corresponding M2(R)-groups, ?21 and ? 2 3 are monogenic under Action 2. (c) Since ?21 contains a type-0 M2(R)-group, T 2 2 , which is not its direct- summand, ?21 ? K(? 2). On the other hand, ?23 has no M2(R)-kernel and each of its proper M2(R)-subgroups are of type-2. Hence, ?23 is of type-s. (d) The classes of monogenic M2(R)-subgroups of ?22 are K(?2) := {(0),?22,? 2 1}; B(? 2) := {T 22 }; G(?2) := {?2,?21,? 2 2,? 2 3, T 2 1 , T 2 3 , T 2 4 , T 2 5 } The socle ideal of M2(R) is therefore given by Soi(M2(R)) = (0 : ?22) ? (0 : ? 2 1). (e) The nil-rigid series of M2(R) is (0) = L1 ? C1 ? L2 ? C2 = M2(R) where L1 = J0(M2(R)) = (0), C1 = Soi(M2(R)) = (0 : ?22) ? (0 : ? 2 1), L2/C1 = J0 (M2(R)/C1) ? 3? j=1 ( 0 : T 2j ) , C2/L2 = Soi (M2(R)/L2) = M2(R)/L2 because K (M2(R)/L2) = ?. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 124 The nil-rigid length of M2(R) is ?M2(R) = 2, hence we have that the s-socle of M2(R), is A ? Soi(M2(R)). By Lemma 5.3.11 A = Soi(M2(R)) ? Js(M2(R)), so the s-socle A is the intersection (0 : ?22) ? (0 : ? 2 1) ? (0 : ? 2 2) ? ( 3? j=1 (0 : (Tj) 2) ) ? ( 2? i=1 (0 : (?i/T2) 2) ) . 3. There is a matrix W in A? which is not in A. (a) Define K := {k0, k1, k2, k3}, where k0 is the zero-map, k1((3, 1, 1)) = (0, 1, 0), k2((3, 1, 1)) = (2, 0, 0), k3((3, 1, 1)) = (2, 1, 0), and if x? negationslash= (3, 1, 1), k1(x?) = k2(x?) = k3(x?) = ? ? ? 0? if x? ? ?1 x? if x? ? ? \ ?1 . Then K is a non-monogenic R-group and it is R-isomorphic to ?1. That is, K is of K2-form. So that K2 = M2(R)?k1, k2? is an M2(R)-group of type-K under Action 2. Hence A ? (0 : K2). (b) Define a matrix W = f s11(f t 11 + f t 12) CHAPTER 5. ?-PRIMITIVITY IN MN(R) 125 where t((0, 1, 0)) = (0, 1, 0), t((2, 0, 0)) = (2, 0, 0), and 0? otherwise, s((2, 1, 0)) = (2, 1, 0), and 0? otherwise. For any ?r, r?? in R2 we have W ?r, r?? = ?s(tr + tr?), 0?. (i) We now show that W negationslash? A ? (0 : K2). Note that s(tr + tr?)(0, 1, 0) ? s(t(T1) + t(T1)) ? s(T1) = (0), (5.3) s(tr + tr?)(2, 0, 0) ? s(t(T2) + t(T2)) ? s(T2) = (0), (5.4) s(tr + tr?)(2, 1, 0) ? s(t(T3) + t(T3)) = s((0)) = (0). (5.5) For the generator ?k1, k2? of K2, we have W ?k1, k2? = ?s(tk1 + tk2), 0?. But s(tk1 + tk2)(3, 1, 1) = s(t((0, 1, 0)) + t((2, 0, 0))) = s((0, 1, 0) + (2, 0, 0)) = s((2, 1, 0)) negationslash= (0). Hence, W negationslash? (0 : K2) and therefore W negationslash? A. (ii) We prove that W ? A?. In order to prove that W ? A?, we need only show W ?r, r?? ? A2 for all r, r? ? R. CHAPTER 5. ?-PRIMITIVITY IN MN(R) 126 Since A = [ 5? j=1 (0 : Tj) ] ? (0 : ?2) ? (0 : ?2), it suffices to show that s(tr + tr?)(x) = 0, (5.6) for any x ? [ 5? j=1 Tj ] ? ?2 ? ?2. If x ? T1 ? T2 ? T3, then by (5.3), (5.4), and (5.5), it follows that (5.6) holds. If x ? T4 ? T5, then s(tr + tr?)x ? s(t(Ti) + t(Ti)) = s((0)) = (0), i = 4, 5. If x ? ?2, then s(tr + tr?)x ? s(t(?2) + t(?2)) ? s(T2) = (0). If x ? ?2, then s(tr + tr?)x ? s(t(?2) + t(?2)) ? s(T1) = (0). Therefore W ? A?. Chapter 6 Nil-rigid series of Mn(R) This chapter focuses on applying the theory of Chapter 3 to matrix near- rings. It is shown how R-groups of ?n-form affect the ?-chains and their lengths. Sufficient conditions under which the nil-rigid series of Mn(R) is longer than that of R are stated. The positions of the s-socle ideals; A+, A? and A, with respect to the (?n)-ideal series, are located. The ()+-series and ()?-series are defined in a natural way. Sufficient conditions under which the ()+-series, ()?-series and the nil-rigid series of Mn(R) are related to the (?n)-ideal series are stated. Examples of near-rings, with their nil-rigid lengths strictly less than that of their associated matrix near-rings are given in Section 6.3. Throughout this chapter R denotes a zero-symmetric right near-ring with identity satisfying the DCCS. The nil-rigid length of R is denoted by ?R. 127 CHAPTER 6. NIL-RIGID SERIES OF MN(R) 128 6.1 The nil-rigid length of Mn(R) In this section, it is shown that, under certain conditions, knowing all ?- chains (up to isomorphism) and where the R-groups of ?n-form appear in the ?-chains, provides sufficient information to determine the nil-rigid length of Mn(R). We refer the reader to Section 3.2 for definitions relating to ?- chains. The following theorem is a consequence of Theorem 4.1.3, Lemma 4.1.7 and the definition of R-groups of ?n-form. Theorem 6.1.1 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Let ? be a faithful locally monogenic R-group, and ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ?? (6.1) be an ?-chain. If ? has no R-subgroups of ?n-form, ? = 0,K, then ?n1 supersetnoteql ? n 2 supersetnoteql ? ? ? supersetnoteql ? n ? is an (?n)-chain. We now generalize this notion. Theorem 6.1.2 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Let ? be a faithful locally monogenic CHAPTER 6. NIL-RIGID SERIES OF MN(R) 129 R-group, and ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ?? (6.2) be an ?-chain. An (?n)-chain corresponding to (6.2) is formed as follows. (a) Let ?1 be an R-subgroup of ? which is maximal amongst the type-0 and 0n-form R-subgroups of ? containing ?1. Then ?n1 is a maximal type-0 Mn(R)-subgroup of ?n containing ?n1 and has ?n1 supersetnoteql ? n 2 . (b) For each odd i, 1 ? i ? ?? 1, let ?i+1 be an R-subgroup of ? which is maximal amongst type-K and Kn-form R-subgroups of ?i containing ?i+1. Then ?ni+1 is a maximal type-K Mn(R)-subgroup of ? n i containing ?ni+1 and has ?ni+1 supersetnoteql ? n i+2, if i+ 2 ? ?. (c) For each even i, 1 ? i ? ?, let ?i+1 be an R-subgroup of ? which is maximal amongst type-0 and 0n-form R-subgroups of ?i containing ?i+1. Then ?ni+1 is a maximal type-0 Mn(R)-subgroup of ? n i containing ? n i+1 and has ?ni+1 supersetnoteql ? n i+2 if i+ 2 ? ?. Note. If ?n? is not of type-s then the above procedure can be continued, making the (?n)-chain in Mn(R) strictly longer than the chain (6.2). Using the above procedure we observe the following. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 130 Theorem 6.1.3 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Let ? be a faithful locally monogenic R-group, and ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ?? (6.3) an ?-chain, where ?1 is not contained in any 0n-form R-subgroup of ?. Also, let H and K be R-subgroups of ? such that H supersetnoteql K. Assume that one of the following cases occurs. (a) There exists an odd i such that H is maximal amongst Kn-form R- subgroups of ? with ?i supersetnoteql H supersetnoteql ?i+1, and there exists K maximal amongst type-0 and 0n-form R-subgroups of ? with H supersetnoteql K supersetnoteql ?i+1. (b) There exists an even i such that H is maximal amongst 0n-form R- subgroups of ? with ?i supersetnoteql H supersetnoteql ?i+1, and there exists K maximal amongst type-K and Kn-form R-subgroups of ? with H supersetnoteql K supersetnoteql ?i+1. Then there is an (?n)-chain corresponding to the ?-chain (6.3) of length at least ? + 2. If the conditions of Theorem 6.1.3 hold and the ?-chain (6.3) has at least CHAPTER 6. NIL-RIGID SERIES OF MN(R) 131 one pair of R-groups, H and K, as described in the above theorem, we say such an ?-chain has a refinement. Theorem 6.1.3 can now be rephrased, under the conditions stated in the theorem, as follows. If the ?-chain, ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ??? , (6.4) has a refinement, then the length, ??n, of its corresponding (?n)-chain sat- isfies ??n ? ?? + 2. By definition, the ?-length of (6.4) is ???12 . Thus the (? n)-length of the corresponding (?n)-chain is ??n ? 1 2 ? (?? + 2)? 1 2 = ( ?? ? 1 2 ) + 1. (6.5) Denote the KG-length of ? and ?n by ?R and ?Mn(R), respectively. The next corollary follows readily. Corollary 6.1.4 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS, and let ? be a faithful locally mono- genic R-group. If a KG-chain of ? has a refinement, then ?Mn(R) ? ?R + 1. Note that a refinement of an ?-chain does not occur if ? has no non- monogenic R-subgroups of ?n-form, ? = 0,K. That is, under this condition, every ?-chain of ?-length l corresponds to an (?n)-chain of (?n)-length l. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 132 Theorem 6.1.5 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS, and let ? be a faithful locally mono- genic R-group. If ? has no R-subgroups of ?n-form, ? = 0,K, then ?Mn(R) = ?R. 6.2 Series of ideals of Mn(R) In this section, the positions of the s-socle ideals; A+, A? and A, with respect to the (?n)-ideal series, are located. The ()+-series, ()?-series are defined in a natural way. Sufficient conditions are given which allow the ()+-series, ()?-series and the nil-rigid series of Mn(R) to be easily related to the (?n)-ideal series. Throughout this section, A denotes the s-socle of R, and A denotes the s-socle of Mn(R). The next proposition relates the s-socle ideal of R to the ?-chains of R. Recall: If ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ?? is a KG-chain then the KG-length, ?R = ??1 2 . That is, ? = 2?R + 1. Thus, if ?R > 0 then ? > 1. Proposition 6.2.1 Let R be a zero-symmetric right near-ring with iden- tity such that Mn(R) satisfies the DCCS, and let ? be a faithful locally monogenic R-group. Let ?1 supersetnoteql ?2 supersetnoteql ? ? ? supersetnoteql ??, (6.6) be a KG-chain. If the KG-length, ?R > 0 and the nil-rigid length of R, CHAPTER 6. NIL-RIGID SERIES OF MN(R) 133 ? = ?+12 , then (0 : ???1) ? A. Moreover, T??1 ? A, where Tj, j = 1, 2, . . . , ?, are ?-ideals as defined in Definition 3.2.11. Proof Consider the ?-ideal series of R, in Definition 3.2.11 and Proposi- tion 3.2.12, T1 ? T2 ? . . . ? T??1 ? T?. Since (6.6) is a KG-chain, (0 : ???1) ? T??1. By Corollary 3.2.17, T??1 ? C??1, where ?? 1 = [? ? 1]/2. Since C??1 ? A, by Lemma 2.3.14, the result follows. square The (?n)-ideal series of Mn(R) is defined in the same way as the ?-ideal series of R was defined in Definition 3.2.11 and Proposition 3.2.12. To avoid confusion, we denote the (?n)-ideal series of Mn(R) by T1 ? T2 ? ? ? ? ? T??1 ? T?. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 134 Theorem 6.2.2 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS, and let ? be a faithful locally mono- genic R-group. If the KG-length of R is ?R > 0, the nil-rigid length ? = ?+12 , and ? has no R-subgroups of ?n-form, ? = 0,K, then A? ? T 2?R . Proof Since ? has no R-subgroups of ?n-form, ? = 0,K, by Theorem 6.1.1 each (?n)-chain is of the form ?nl,1 supersetnoteql ? n l,2 supersetnoteql ? ? ? supersetnoteql ? n l,?(l) corresponding to ?l,1 supersetnoteql ?l,2 supersetnoteql ? ? ? supersetnoteql ?l,?(l), where l ? ? indexes the ?-chains, as in Definition 3.2.11. Thus the j-th ?-ideal Tj has T ?j = ? ? ? ? l?? ?(l)?j (0 : ?l, j) ? ? ? ? = ? l?? ?(l)?j (0 : ?l, j) ?. Since (0 : ?l, j) ? ? (0 : ?nl, j), by Lemma 4.2.5, it follows that T ???1 ? T??1. But A ? T??1, by Propo- sition 6.2.1, and hence A? ? T ???1, by Proposition 2.4.6(i). Therefore A? ? T??1. This proves the theorem since ?R = ??1 2 . square Corollary 6.2.3 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Let ? be a faithful locally monogenic CHAPTER 6. NIL-RIGID SERIES OF MN(R) 135 R-group, and the KG-length of R be ?R > 0 with the nil-rigid length ? = ?+12 . If ? has no R-subgroups of ?n-form, ? = 0,K, then A+ ? T 2?R . Denote the nil-rigid series of Mn(R) by L1 subsetnoteql C1 subsetnoteql L2 subsetnoteql C2 subsetnoteql ? ? ? subsetnoteql L?Mn(R) subsetnoteql C?Mn(R) , and the nil-rigid length of Mn(R) by ?Mn(R) to avoid ambiguity. The following lemma and theorems follow as direct application of Theorems 3.2.7, 3.2.16 and 3.2.17, in terms of matrix near-rings. Lemma 6.2.4 Let Mn(R) be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, and having the nil-rigid length, ?Mn(R) greaternotequal 3. Sup- pose that ?n is a faithful Mn(R)-group with KG-length ?. Let ?i, 1 supersetnoteql ?i, 2 supersetnoteql . . . supersetnoteql ?i, ??1 supersetnoteql ?i, ?, be a KG-chain of ?n. (a) If j is even then ?i,j is a type-K ( Mn(R)/Lj/2 ) -group. (b) If j is odd then ?i,j is a type-0 ( Mn(R)/C(j?1)/2 ) -group. Theorem 6.2.5 Let Mn(R) be a zero-symmetric right near-ring with iden- tity satisfying the DCCS, and ?n be a faithful Mn(R)-group. (a) If j is even then Tj/L(j/2) ? Soi ( Mn(R)/L(j/2) ) . CHAPTER 6. NIL-RIGID SERIES OF MN(R) 136 (b) If j is odd then Tj/C(j?1)/2 ? J0 ( Mn(R)/C(j?1)/2 ) . Here Tj is the j-th (?n)-ideal of Mn(R), and L(j/2) and C(j?1)/2 are ideals in the nil-rigid series of Mn(R). Theorem 6.2.6 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS, and let ? be a faithful locally mono- genic R-group. If ? has no R-subgroups of ?n-form, ? = 0,K, then the (?n)-ideal series of Mn(R), T1 ? T2 ? ? ? ? ? T??1 ? T?, and the nil-rigid series of Mn(R), L1 subsetnoteql C1 subsetnoteql ? ? ? subsetnoteql L?Mn(R)?1 subsetnoteql C?Mn(R)?1 subsetnoteql L?Mn(R) subsetnoteql C?Mn(R) , are related by T2i?1 ? Li and T2i ? Ci, where 1 ? i ? ?Mn(R). If a faithful R-group, ?, has no R-subgroups of ?n-form, ? = 0,K, it follows from Theorem 6.2.6 that C??1 ? T2(??1). By definition of the s-socle, A ? C??1, see Lemma 2.3.14, hence the next theorem. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 137 Theorem 6.2.7 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS, and let ? be a faithful locally mono- genic R-group. Let the KG-length of R be ?R > 0. If ? has no R- subgroups of ?n-form, ? = 0,K, then A ? T2(??1). Definition 6.2.8 For the nil-rigid series of R; (0) ? L1 subsetnoteql C1 subsetnoteql ? ? ? subsetnoteql L??1 subsetnoteql C??1 subsetnoteql L? subsetnoteql C? = R, (a) the series of ideals of Mn(R), (0) ? L+1 subsetnoteql C + 1 subsetnoteql ? ? ? subsetnoteql L + ??1 subsetnoteql C + ??1 subsetnoteql L + ? subsetnoteql C + ? = Mn(R), will be called the ( )+-series, and (b) the series of ideals of Mn(R), (0) ? L?1 subsetnoteql C ? 1 subsetnoteql ? ? ? subsetnoteql L ? ??1 subsetnoteql C ? ??1 subsetnoteql L ? ? subsetnoteql C ? ? = Mn(R), will be called the ( )?-series. We observe that, just as the annihilator ideal P(?) := ? R??? (0? : (R?)n) contains (J?(R)) +, J?(Mn(R)) and (J?(R))?, for ? = 0, s, 2, the (?n)-ideal series contains the ()+-series, the nil-rigid series, and the ()?-series, when R has no R-subgroups of ?n-form. This fact is recorded in the next theorem. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 138 Theorem 6.2.9 Let R be a zero-symmetric right near-ring with identity such that Mn(R) satisfies the DCCS. Let ? be a faithful locally monogenic R-group, and (0) ? L1 subsetnoteql C1 subsetnoteql ? ? ? subsetnoteql L??1 subsetnoteql C??1 subsetnoteql L? subsetnoteql C? = R, be the nil-rigid series of R. If ? has no R-subgroups of ?n-form, ? = 0,K, then the (?n)-ideal series of Mn(R), T1 ? T2 ? ? ? ? ? T??1 ? T?, contains the ()+-series, the ()?-series and the nil-rigid series of Mn(R), in the sense of Theorem 6.2.6. 6.3 Examples on the nil-rigid length of Mn(R) In this section, examples of near-rings, with their nil-rigid lengths strictly less than that of their associated matrix near-rings are given. The symbol ?G is used denote group theoretic direct sum. Throughout this section, let ? := Z32 ?G Z2. Consider the following subgroups of the group Z32; H1 = {0, 2, 4, . . . , 28, 30}, H2 = {0, 4, 8, 12, 16, 20, 24, 28}, CHAPTER 6. NIL-RIGID SERIES OF MN(R) 139 H3 = {0, 8, 16, 24} and H4 = {0, 16}. Define the following subgroups of ?; ?1 := H1 ?G Z2, ?2 := H2 ?G Z2, ?3 := H3 ?G Z2, ?1 := H4 ?G Z2, ?2 := H3 ?G {0} = {(0, 0), (8, 0), (16, 0), (24, 0)}, ?3 := {(0, 0), (8, 1), (16, 0), (24, 1)}, S1 := H4 ?G {0} = {(0, 0), (16, 0)}, S2 := {0} ?G Z2 = {(0, 0), (0, 1)}, S3 := {(0, 0), (16, 1)}. We now use ? to give an example which illustrates the fact that ?R may be strictly less than ?Mn(R). Let the near-ring in this example be R ?. We will observe that J0(R?) = (0) so that the socle ideal, Soi(R?) negationslash= (0), is a member of the nil-rigid series. Here, the faithful R?-group ? is a member of a non-trivial ?-chain. Example 6.3.1 Define the near-ring R? by, R? := { f ? M0(?) | f(?i) ? ?i, i = 1, 2, 3; f(?j) ? ?j, j = 1, 2, 3; f(Sl) ? Sl, l = 1, 2; ??, ?? ? ?1, ? ? ?? ? ?2 ? f(?)? f(??) ? ?2 ; ??1, ?2 ? ?2, ?1 ? ?2 ? ?3 ? f(?1)? f(?2) ? ?3 ; ? ?1, ?2 ? ?1, ?1 ? ?2 ? S1 ? f(?1)? f(?2) ? S1 }. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 140 Then R? is a near-ring with multiplicative identity under point-wise addition and map composition. We observe the following. (a) The group ? is a faithful R?-group of type-0. (b) The subgroup ?2 is an R?-kernel of ?1, and ?3 is an R?-kernel of ?2, and ?1/?2 and ?2/?3 are R?-groups of type-2. In addition, both ?1 and ?2 are of type-K. (c) The subgroup ?3 = ?2 ?G S2 is the only non-monogenic R?-subgroup of ?. In fact, ?3 is an R?-group of 02-form. (d) The subgroup ?1 is an R?-group of K2-form but has S1 as an R?- kernel. Thus ?1/S1 is an R?-group of type-2. (e) The R?-groups of type-0 are ?2, ?3, S1 and S2. There is only one non-trivial ?-chain, up to isomorphism, ? supersetnoteql ?1 supersetnoteql ?3. Thus the KG-length of ? is ?R? = 1. Also, we observe that the nil-rigid length of R? is ?R? = 2. Now consider the faithful M2(R?)-group, ?2. Note that ?23 is an M2(R ?)- group of type-0. The chain of M2(R?)-groups corresponding to ? supersetnoteql ?1 supersetnoteql ?3 is ?2 supersetnoteql ?21 supersetnoteql ? 2 3. (6.7) Here, ?21 is maximal type-K M2(R ?)-subgroup of ?2, but (6.7) is not an (?2)-chain because ?23 is not maximal amongst type-0 M2(R ?)-subgroups of CHAPTER 6. NIL-RIGID SERIES OF MN(R) 141 ?21. Therefore, ? 2 has an (?2)-chain, ?2 supersetnoteql ?21 supersetnoteql ? 2 3 supersetnoteql ? 2 1 supersetnoteql S 2 2 , which is the only (?2)-chain, up to isomorphism, hence a KG-chain. We conclude, by Theorem 3.2.7, that the nil-rigid length of M2(R?) is ?M2(R?) ? 3, giving ?Mn(R?) greaternotequal ?R? . The next example involves a refinement of a KG-chain. The near-ring, R, in this example is constructed such that J0(R) negationslash= (0) and Soi(R) = (0). Also, ? is not a member of any ?-chain. Example 6.3.2 Define the near-ring R by, R := { f ? M0(?) | f(?i) ? ?i, i = 1, 2, 3; f(?j) ? ?j, j = 1, 2, 3; f(Sl) ? Sl, l = 1, 2, 3; ??, ?? ? ?, ? ? ?? ? ?1 ? f(?)? f(??) ? ?1 ; ??1, ?2 ? ?2, ?1 ? ?2 ? ?3 ? f(?1)? f(?2) ? ?3 ; ? ?1, ?2 ? ?1, ?1 ? ?2 ? S1 ? f(?1)? f(?2) ? S1 }. Then R is a near-ring with multiplicative identity under point-wise addition and map composition. Observe the following. (a) The group ? is a faithful R-group of type-K, because it has the type-0 R-subgroup ?1 as an R-kernel. Therefore J0(R) negationslash= (0). (b) The R-group ?3 is an R-kernel of ?2, and S1 is an R-kernel of ?1. Therefore, ?2 and ?1 are of type-K. CHAPTER 6. NIL-RIGID SERIES OF MN(R) 142 (c) The R-group ?3 = ?2 ?G S2 is the only non-monogenic R-subgroup of ?. In fact, ?3 is an R-group of 02-form. (d) Since all subgroups of ?1 are R-groups, ?1 is non-monogenic and of K2-form. (e) The type-0 R-subgroups of ? are ?1, ?2, ?3, S1 and S2. There is only one non-trivial ?-chain, which is thus a KG-chain, that is ?1 supersetnoteql ?2 supersetnoteql ?3. (6.8) The KG-length of R is ?R = 1. Hence, the nil-rigid length of R is ?R = 2. Consider ?2 as a faithful M2(R)-group. Note that ?3 and ?1 give a refinement of (6.8), where ?3 is of 02-form and ?1 is of K2-form. An (?2)-chain corresponding to ?1 supersetnoteql ?2 supersetnoteql ?3 is ?21 supersetnoteql ? 2 2 supersetnoteql ? 2 3 supersetnoteql ? 2 1 supersetnoteql S 2 2 , which is the only (?2)-chain, up to isomorphism, and hence a KG-chain of ?2. Therefore, ?M2(R) = 2, and by Theorem 3.2.7, the nil-rigid length of M2(R) is ?M2(R) ? 3. Thus ?Mn(R) greaternotequal ?R. Chapter 7 Conclusion 7.1 A brief summary of results A special type of monogenic R-groups, called R-groups of type-K, was in- troduced. The socle ideal was then characterized as an intersection of anni- hilators of R-groups of type-K. It has been shown that, for near-rings satisfying the DCCL, if R = Soi(R) ? L, then (Soi(R))? is the annihilating ideal of Ln. That is, (Soi(R))? = (0 : Ln). 143 CHAPTER 7. CONCLUSION 144 It was also proved that the 0-radical, J0(R), and the s-radical, Js(R), are distinct if, and only if, there is an R-group of type-K contained in a type-0 R-group. Alternating chains of type-0 and type-K R-subgroups, called ?-chains, of a faithful R-group ?, satisfying the DCCS, were defined. The maximal length of the ?-chains is closely related to the nil-rigid length. Non-monogenic R-groups, whose corresponding Mn(R)-groups are mono- genic, are classified into three ?n-forms, ? = 0, s,K. It is shown that a monogenic R-group is of type-K if, and only if, its corresponding Mn(R)- group is of type-K. The cognizance of R-groups of ?n-form, and where they fit into ?-chains, enables one to give a lower bound on the nil-rigid length of Mn(R), without having to first calculate each ideal in the series. For a zero-symmetric near-ring with identity, and ? a faithful R-group, let I? = ? ?i ?E?(R) (0 : ?i), P(?) = ? ?i ?E?(R) (0 : ?ni ), where E?(R) is the class of all type-? R-subgroups of ?, ? = 0, s, 2,K, and I? = ? ?j ?E?(Mn(R)) (0 : ?j), where E?(Mn(R)) is the class of all type-?, ? = 0, s, 2,K, Mn(R)-subgroups of ?2. Then the following relationships were shown, I+? ? P(?), I ? ? ? P(?), I? ? P(?). CHAPTER 7. CONCLUSION 145 These ideals, which are intersections of annihilators of monogenic R-groups or Mn(R)-groups, are referred to as annihilator ideals. Using this concept, the following relationships were proved, Soi(Mn(R)) ? (Soi(R))?; (Js(R))+ ? Js(Mn(R)) where R satisfies the DCCL. Although the s-socle is not an annihilator ideal, it was shown that A+ ? A, where A and A are the s-socle ideals of R and Mn(R), respectively. A counter-example to a conjecture by Meldrum and Meyer in [17] is given. That is, a near-ring R with (J0(R)) + negationslash? J0(Mn(R)), is constructed. Several examples are presented illustrating the lack of relationships, (Soi(R))+ negationslash? Soi(Mn(R)); A? negationslash? A, and A negationslash? A?, where A and A are s-socles of R and Mn(R), respectively. We conclude that the only annihilator ideal which strictly satisfies the rela- tionship I+? ? I? ? I ? ? CHAPTER 7. CONCLUSION 146 is the Jacobson s-radical. The Jacobson 2-radical fails simply because J2(Mn(R)) = (J2(R))?. That is, a strict inclusion never occurs. Other examples illustrate, (a) a typical R-group of type-K , and a typical R-group of type-0 but not of type-s, (b) how the KG-length relates to the nil-rigid length, (c) that the nil-rigid length of a near-ring can be strictly less than the nil-rigid length of its associated matrix near-rings. We concluded that, besides the lack of one distributive law, or the lack of com- mutativity on the additive group (R,+), the discrepancies between ideals of a near-ring and the corresponding ideals of the associated matrix near-rings are ascribable to the existence of R-groups of ?n-form (? = 0, s,K). 7.2 Some Open Questions Several questions of interest still remain unanswered. 1. What are the necessary and sufficient conditions for an annihilator ideal I? , ? = 0, s, 2,K, of R to be such that I? = I ? ? , (7.1) where I? is the ideal of Mn(R) corresponding to I?? CHAPTER 7. CONCLUSION 147 The condition that every R-subgroup of a faithful R-group must be mono- genic turns out to be too strong. In particular, the Jacobson 2-radical sat- isfies (7.1) regardless of whether every R-group of R is monogenic or not. 2. What are the relationships between any two of the quasi-radicals, (J1/2(R)) +, J1/2(Mn(R)) and (J1/2(R))? ? There is a close connection between the left ideal J1/2(R) and the s-radical, Js(R). For example, it is well known that, (a) for a near-ring R satisfying the DCCL, J1/2(R) is a two-sided ideal if, and only if, J0(R) = Js(R), cf. [4, Theorem 2.12]; (b) For a near-ring satisfying the DCCS and having nil-rigid length ? > 1, Js(R) = J1/2(R) + A + B, where A is the s-socle of R, B = Js(R) ? L??1, and L??1 is the member of the nil-rigid series of R as in Theorem 2.3.19. 3. When is the s-socle an annihilator ideal? This can be rephrased as: when is the s-socle, A, equal to the intersection Js(R) ? C??1, where the nil-rigid length of R is ? > 2? 4. Is there an example of a near-ring R with J0(Mn(R)) negationslash= J1/2(Mn(R)) negationslash= Js(Mn(R)) negationslash= J2(Mn(R)) while J0(R) = J1/2(R) = Js(R) = J2(R)? CHAPTER 7. CONCLUSION 148 5. What conditions are needed on the near-ring, R, for ?R = ?R + 1? 6. Can the theory presented here be extended to other radicals? 7. By considering a group near-ring, R[G], as a subnear-ring of a matrix near-ring Mn(R), if |G| = n, see Meyer [22], can the theory developed here be transfered to the context of group near-rings? Appendix A Action of Mn(R) on factor-groups Consider the group, (?/?)n, where ? = R? is a monogenic R-group and ? an R-kernel of ?. Let ?r1? + ?, r2? + ?, . . . , rn? + ?? ? (?/?) n, and U ? Mn(R). Define the following action of Mn(R) on (?/?)n as U?r1?+?, r2?+?, . . . , rn?+?? := ?s1?+?, s2?+?, . . . , sn?+??, (A.1) where U?r1, r2, . . . , rn? = ?s1, s2, . . . , sn?, 149 APPENDIX A. ACTION OF MN(R) ON FACTOR-GROUPS 150 for some si ? R, 1 ? i ? n. To show that action (A.1) is well-defined, suppose ?r1? + ?, r2? + ?, . . . , rn? + ?? = ?a1? + ?, a2? + ?, . . . , an? + ??, for ri, ai ? R, 1 ? i ? n. It follows that ri? + ? = ai? + ?, 1 ? i ? n. But, for any r ? R, we have rri? + ? = rai? + ?, hence rri? ? rai? = (rri ? rai)? ? ?, 1 ? i ? n. (A.2) We now prove, by induction on the weight of U , that the action (A.1) is well-defined. Let U = f rij, then U?r1, r2, . . . , rn? = ?0, . . . , 0, rrj, 0, . . . , 0? and U?a1, a2, . . . , an? = ?0, . . . , 0, raj, 0, . . . , 0?, where both rrj and raj appear in i-th places. Thus U?r1? + ?, r2? + ?, . . . , rn? + ?? = ??, . . . ,?, rrj? + ?,?, . . . ,?? and U?a1? + ?, a2? + ?, . . . , an? + ?? = ??, . . . ,?, raj? + ?,?, . . . ,?? where both rrj? + ? and raj? + ? appear in i-th places. Hence, by (A.2), U?r1? + ?, r2? + ?, . . . , rn? + ?? ? U?a1? + ?, a2? + ?, . . . , an? + ?? APPENDIX A. ACTION OF MN(R) ON FACTOR-GROUPS 151 = ??, . . . , ?, (rrj ? raj)? + ?, ?, . . . , ?? = ??,?, . . . ,??. Let U = f b1ij + f b2 i l , then U?r1, r2, . . . , rn? = ?0, . . . , 0, b1rj + b2rl, 0, . . . , 0? and U?a1, a2, . . . , an? = ?0, . . . , 0, b1aj + b2al, 0, . . . , 0?, where both b1rj + b2rl and b1aj + b2al appear in i-th places. Hence U?r1?+?, r2?+?, . . . , rn?+?? = ??, . . . ,?, (b1rj+b2rl)?+?,?, . . . ,?? and U?a1?+?, a2?+?, . . . , an?+?? = ??, . . . ,?, (b1aj+b2al)?+?,?, . . . ,??, where both (b1rj + b2rl)? + ? and (b1aj + b2al)? + ? appear in i-th places. It now follows by (A.2) and the first step of induction that U?r1? + ?, r2? + ?, . . . , rn? + ?? ? U?a1? + ?, a2? + ?, . . . , an? + ?? = ??, . . . , ?, ( (b1rj+b2rl)?(b1aj+b2al) ) ?+?, ?, . . . , ?? = ??, . . . , ?, ( b1rj+b2rl?b2al?b1aj ) ?+?, ?, . . . , ?? = ??,?, . . . ,??. Assume that the action (A.1) of Mn(R) on (?/?)n is well-defined for any matrix of weight less than m, where m is a positive integer. APPENDIX A. ACTION OF MN(R) ON FACTOR-GROUPS 152 Let V and W be matrices each of weight less than m, and write V ?r1, r2, . . . , rn? = ?r?1, r ? 2, . . . , r ? n?, V ?a1, a2, . . . , an? = ?a ? 1, a ? 2, . . . , a ? n?, W ?r?1, r ? 2, . . . , r ? n? = ?r ?? 1 , r ?? 2 , . . . , r ?? n? and W ?a ? 1, a ? 2, . . . , a ? n? = ?a ?? 1, a ?? 2, . . . , a ?? n?. Suppose U = V +W . It follows from by the induction hypothesis that U?r1? + ?, r2? + ?, . . . , rn? + ?? ? U?a1? + ?, a2? + ?, . . . , an? + ?? = ( ?r?1? + ?, r ? 2? + ?, . . . , r ? n? + ?? + ?r ?? 1? + ?, r ?? 2? + ?, . . . , r ?? n? + ?? ) ? ( ?a?1? + ?, a ? 2? + ?, . . . , a ? n? + ?? + ?a ?? 1? + ?, a ?? 2? + ?, . . . , a ?? n? + ?? ) = ?(r?1+r ?? 1?a ?? 1?a ? 1)?+?, (r ? 2+r ?? 2?a ?? 2?a ? 2)?+?, . . . , (r ? n+r ?? n?a ?? n?a ? n)?+?? = ?(r?1 ? a ? 1)? +?, (r ? 2 ? a ? 2)? +?, . . . , (r ? n ? a ? n)? +?? = ??,?, . . . ,??. Now suppose U = WV . It follows again by the induction hypothesis that U?r1? + ?, r2? + ?, . . . , rn? + ?? ? U?a1? + ?, a2? + ?, . . . , an? + ?? = W ?r?1?+?, r ? 2?+?, . . . , r ? n?+?? ? W ?a ? 1?+?, a ? 2?+?, . . . , a ? n?+?? = ?r??1?+?, r ?? 2?+?, . . . , r ?? n?+?? ? ?a ?? 1?+?, a ?? 2?+?, . . . , a ?? n?+?? = ?(r??1?a ?? 1)?+?, (r ?? 2?a ?? 2)?+?, . . . , (r ?? n?a ?? n)?+?? = ??,?, . . . ,??. Therefore action (A.1) is well-defined. square Appendix B Structure of ideals of the form I+ Let I and J be non-trivial ideals of a zero-symmetric right near-ring, R, with identity, such that J ? I. The zero-element of (J/I)+ subsetnoteql Mn(R/I) is denoted by 0. Let x, xij, z ? J , i, j = 1, 2, . . . , n, r ? R, and consider the map ? : J+ ? (J/I)+ defined recursively by n? i,j=1 fxijij mapsto? n? i,j=1 f (xij + I)ij , 153 APPENDIX B. STRUCTURE OF IDEALS OF THE FORM I+ 154 and f rp? ( n? i,j=1 fxijij ) mapsto? f (r+ I)p? ( n? i,j=1 f (xij + I)ij ) . Note. By right distributivity, ( n? p,q=1 f rpqpq )( n? i,j=1 f rijij ) = n? p,q=1 ( f rpqpq n? i,j=1 f rijij ) . (B.1) We now give some properties of the generating set, { faij | a ? I, 1 ? i, j ? n }, of I+, in line with Section E in Chapter 1 of [19]. (a) For a ? I and r ? R with a+ r ? I, we note that faij + f r ij ? I +, however, if p negationslash= i or ? negationslash= j, then fap? + f r ij ? I +, only if r ? I. (b) For a, b ? I and r ? R with a+ r ? b ? I, we note that faij + f r ij ? f b ij ? I +, but if p negationslash= i negationslash= l or ? negationslash= j negationslash= q, then fap? + f r ij ? f b lq ? I +, APPENDIX B. STRUCTURE OF IDEALS OF THE FORM I+ 155 only if r ? I. (c) For a ? I and r, r? ? R with r(r? + a)? rr? ? I, we note that f r(r ?+a) ij + f rr? ij ? I +, whereas, if p negationslash= i or ? negationslash= j, then f r(r ?+a) p? + f rr? ij ? I +, only if r ? I or r? ? I. (d) For any ?? = ?r1, r2, . . . , rn?, r, r? ? R and z ? J , let U := f rp? ( f r ? ij + f z lq ) ? f rp?f r? ij , then U?? = ? ?????????? ?????????? ?0, 0, . . . , 0?, if ? negationslash= i and ? negationslash= l ?0, . . . , 0, rr?rj ? rr?rj, 0, . . . , 0?, if ? = i and ? negationslash= l ?0, . . . , 0, rzrq, 0, . . . , 0?, if ? negationslash= i and ? = l ?0, . . . , r(r?rj + zrq)? r(r?rj), 0, . . . , 0?, if ? = i = l and j negationslash= q ?0, . . . , r(r? + z)rj ? rr?rj, 0, . . . , 0?, if ? = i = l and j = q. Now U can be written in terms of the generators, { faij | a ? R, 1 ? i, j ? n }, of Mn(R) as U = ? ?????????? ?????????? f 0ij, if ? negationslash= i and ? negationslash= l f 0ij, if ? = i and ? negationslash= l f rzpq , if ? negationslash= i and ? = l f rp? ( f r ? ?j + f z ?q ) ? f rp?f r? ?j, if ? = i = l and j negationslash= q f r(r ?+z)?rr? pq , if ? = i = l and j = q. Note that, in the fourth case, i.e., U = f rp? ( f r ? ?j + f z ?q ) ? f rp?f r? ?j, we have U?? = ?0, . . . , r(r?rj + zrq)? r(r ?rj), 0, . . . , 0?. APPENDIX B. STRUCTURE OF IDEALS OF THE FORM I+ 156 If U ? J+ ? I? then r(r?rj + zrq) ? rr? ? I, for all rj, rq ? R. In particular, for rj = 1R = rq, we have r(r?1R + z1R)? r(r ?1R) = r(r ? + z)? rr? ? I. Hence, U = f rp? ( f r ? ?j + f z ?q ) ? f rp?f r? ?j ? I +. Thus, in general, if U := f rp? ( f r ? ij + f z lq ) ? f rp?f r? ij ? J +? I?, then U ? I+. As in Section E of Chapter 1 of [19], the ideal, J+ = Id{ f?ij |? ? J, 1 ? i, j ? n}, is constructed by the following recursive rules: (i) faij ? J +, for all a ? J ; (ii) If faij, f b lq ? J +, then faij + f b lq ? J +; (iii) If faij ? J + and f rp? ? Mn(R), then f a ijf r p? ? J +; (iv) If faij ? J + and f rp? ? Mn(R), then f a ij + f r p? ? f a ij ? J +; (v) If faij ? J + and f rp?, f r? lq ? Mn(R), then f rp? ( f r ? ij + f a lq ) ? f rp?f r? ij ? J +; (vi) Nothing else is in J+. Proposition B.0.1 Let R be a zero-symmetric right near-ring with iden- tity, and with non-trivial ideals I and J such that J ? I. Let ? be as defined above, then ker(?) ? I+. Proof Let U be an element of ker(?) = J+ ? I?. That is, U is an APPENDIX B. STRUCTURE OF IDEALS OF THE FORM I+ 157 matrix in Mn(R) such that ?(U) = 0. We prove the result by induction on the weight of U . Let U = fxij, then ?(U) = f (x+I)ij = f (0+I) ij = 0, which implies x ? I and hence U ? I+. Let U = fxij + f z lq, then ?(U) = f (x+I)ij + f (z+I) lq = 0. If i = l and j = q, then ?(U) = f (x+z)+Iij = 0, hence x + z ? I, and U ? I+. If i negationslash= l or j negationslash= q, then ?(U) = f (x+I)ij +f (z+I) lq = f (0+I) ij +f (0+I) lq . Thus, x ? I and z ? I, hence U ? I+. Let U = f rp? ( f r ? ij + f z lq ) ? f rp?f r? ij , then ?(U) = ? ?????????? ?????????? f (0+I)ij , if ? negationslash= i and ? negationslash= l f (0+I)ij , if ? = i and ? negationslash= l f (rz+I)pq , if ? negationslash= i and ? = l f (r+I)p? ( f (r ?+I) ?j + f (z+I) ?q ) ? f (r+I)p? f (r?+I) ?j , if ? = i = l and j negationslash= q f (r(x+z)+I)pq , if ? = i = l and j = q. In the fourth case of ?(U), we have f (r+I)p? ( f (r ?+I) ?j + f (z+I) ?q ) ? f (r+I)p? f (r?+I) ?j = 0, only if (r+I) ( (r?+I)+(z+I) ) ?(r+I)(r?+I) = ( r(r?+z)?rr? ) +I = 0+I = 0. APPENDIX B. STRUCTURE OF IDEALS OF THE FORM I+ 158 That is, r(r? + z)? rr? ? I, in which case, U ? I+. Thus, for all values of the indices, p, ?, i, j, l, and q, we have that U = f rp? ( f r ? ij + f z lq ) ? f rp?f r? ij ? ker(?) implies U ? I +. Assume that, for any matrix, U ? Mn(R), of weight less than m, where m is a positive integer, if U ? ker(?), then U ? I+. Let U and V be matrices in ker(?), each of weight less than m. By the induction hypothesis, U ? I+ and V ? I+. It follows by the way in which I+ is generated that U + V ? I+ and UV ? I+. square Bibliography [1] J. C. Beidleman. On the theory of radicals of distributively generated near-rings. Math. Annalen, 173:89?101, 1967. [2] D. W. Blackett. Simple and semisimple near-rings. Proc. Amer. Math. Soc., 4:772?785, 1953. [3] J. F. T. Hartney. Radicals and anti-radicals of near-rings. Doctoral dissertation, University of Nottingham, 1979. [4] J. F. T. Hartney. A radical for near-rings. Proc. Royal Soc. Edin., 93A:105?110, 1982. [5] J. F. T. Hartney. An anti-radical for near-rings. Proc. Royal Soc.Edin., 96A:185?191, 1984. [6] J. F. T. Hartney. On the decomposition of the s-radical of a near-ring. Proc. Edin. Math. Soc., 33:11?22, 1990. [7] J. F. T. Hartney. s-Primitivity in matrix near-rings. Quaestiones Math., 18:487?500, 1995. 159 BIBLIOGRAPHY 160 [8] J. F. T. Hartney and A. M. Matlala. On the nilpotence of the s-radical in matrix near-rings. In Near-rings and Near-fields, pages 217?224. Springer, 2005. [9] J. F. T. Hartney and S. N. Mavhungu. s-Primitive ideals in matrix near- rings. In Near-rings and Near-fields, pages 103?107. Kluwer Academic Publishers, 2000. [10] H. E. Heatherly. Matrix near-rings. Journal London Math. Soc., (2) 7:355?356, 1973. [11] R. R. Laxton. Prime ideals and the ideal-radical of a distributively generated near-ring. Math. Zeitschr., 83:8?17, 1964. [12] R. R. Laxton and A. Machin. On the decomposition of near-rings. Abh. Math. Sem. University of Hamburg, 38:221?230, 1972. [13] S. Ligh. A note on matrix near-rings. Journal London Math. Soc., (2) 11:383?384, 1975. [14] J. D. P. Meldrum. Near-rings and their links with groups. Number 134 in Research Notes in Mathematics. Pitman Advanced Publishing Program, London, 1985. [15] J. D. P. Meldrum and J. H. Meyer. Modules over matrix near-rings and the J0-radical. Monatsh. Math., 112:125?139, 1991. [16] J. D. P. Meldrum and J. H. Meyer. Intermediate ideals in matrix near- rings. Communications in Algebra, 24:1601?1619, 1996. BIBLIOGRAPHY 161 [17] J. D. P. Meldrum and J. H. Meyer. The J0-radical of a matrix near-ring can be intermediate. Canad. Math. Bull., 40(2):198?203, 1997. [18] J. D. P. Meldrum and A. P. J. van der Walt. Matrix near-rings. Archiv. Math., 47:312?319, 1986. [19] J. H. Meyer. Matrix near-rings. Doctoral dissertation, University of Stellenbosch, 1986. [20] J. H. Meyer. Left ideals and 0-primitivity in matrix near-rings. Proc. Edin. Math. Soc., 35:173?187, 1992. [21] J. H. Meyer. Chains of intermediate ideals in matrix near-rings. Archiv Math., 63:311?315, 1994. [22] J. H. Meyer. Two-sided ideals in group near-rings. Journal Austral. Math. Soc., 77:321 ? 334, 2004. [23] G. Pilz. Near-Rings: The Theory and its Applications. Number 23 in North-Holland Mathematics Studies. North-Holland Publishing Com- pany, Amsterdam, 1977. [24] D. Ramakotaiah. Radicals for near-rings. Math. Zeitsch., 97:45?56, 1967. [25] S. D. Scott. Formation radicals of near-rings. Proc. London Math. Soc., 25:441?464, 1972. [26] A. P. J. van der Walt. Primitivity in matrix near-rings. Quaestiones Math., 9:459?469, 1986. BIBLIOGRAPHY 162 [27] A. P. J. van der Walt. On two-sided ideals in matrix near-rings. In Near-rings and Near-fields, (ed, G.Betsch) North Holland, Amsterdam, pages 267?271, 1987. Index ()?-series, 137 ()+-series, 137 (?n)-chain, 128 (?n)-ideal series, 133 0n-form, 81 ?-chain, 55 ?-chain refinement, 131 ?-ideal series, 61 ?-length, 55 F-decomposable, 24 F-decomposition, 24 KG-chain, 55, 59 KG-length, 59 Kn-form, 82 ?-maximal, 22 ?-primitive, 21 ?-radical, 21 s-socle, 28 sn-form, 82 Action 2, 80 annihilating ideal, 18 annihilator ideal, 73 critical ideal, 29 Crux, 25 factor R-group, 16 faithful R-group, 18 homomorphism, 16 intermediate ideal, 34 Jacobson-type radical, 21 locally monogenic, 74 matrix near-ring, 31 modular, 22 ?-modular, 22 monogenic, 15 near-ring, 14 nil-radical, 25 nil-rigid length, 26, 58 163 INDEX 164 nil-rigid series, 26 quotient near-ring, 18 R-group, 15 R-kernel, 16 R-subgroup, 15 socle-decomposition, 25 socle-ideal, 24 subfactor, 29 type-K, 39 type-0, 19 type-2, 19 type-s, 19 weight of a matrix, 33 zero-symmetric, 14