RIGHT DISTRIBUTIVELY GENERATED NEAR-RINGS AND THEIR LEFT/RIGHT REPRESENTATIONS

Abstract

For right near-rings the left representation has always been considered the natural one. A study of right representation for right distributively generated (d.g.) near-rings was initiated by Rahbari and this work is extended here to introduce radical-like objects in the near-ring R using right R-groups. The right radicals rJ0(R), rJ1/2(R) and rJ2(R) are defined as counterparts of the left radicals J0(R), J1/2(R) and J2(R) respectively, and their properties are discussed. Of particular interest are the relationships between the left and right radicals. It is shown for example that for all finite d.g. near-rings R with identity, J2(R) = rJ0(R) = rJ1/2(R) = rJ2(R). A right anti-radical, rSoi(R), is defined for d.g. near-rings with identity, using a construction that is analogous to that of the (left) socle-ideal, Soi(R). In particular, it is shown that for finite d.g. near-rings with identity, an ideal A is contained in rSoi(R) if and only if A \ J2(R) = (0). The relationship between the left and right socle-ideals is investigated, and it is established that rSoi(R)  Soi(R) for d.g. near-rings with identity and satisfying the descending chain condition for left R-subgroups.