Units of compatible nearrings, IV

Abstract

This is the fourth in a sequence of papers originating in a effort to study the units of a compatible nearring \(R\) satisfying the descending chain condition on right ideals using a faithful compatible module \(G\) of \(R\). A key point in this endeavor involves determining \(1 + Ann_R(G/H)\) where \(H\) is a direct sum of isomorphic minimal \(R\)-ideals where success in doing so gives us not only information about the units of \(R\), but also information about \(R\) and \(J_2(R)\). In the previous papers, \(1 + Ann_R(G/H)\) has been determined whenever \(G/H\) does not contain a minimal factor isomorphic to the minimal summands of \(H\). In this paper we determine \(1 + Ann_R(G/H)\) when \(G/H\) does contain a minimal factor isomorphic to the minimal summands of \(H\). With the completion of the determination of \(1 + Ann_R(G/H)\) in all cases, we illustrate how things work in practice by considering the nearrings generated by the inner automorphisms of a finite dihedral group, special linear group, and general linear group and nearrings of congruence preserving functions on an expanded group.