Radicals and anti-radicals of near-rings and matrix near-rings

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 nearring 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 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.