Centralizer near-rings that are endomorphism rings

Abstract

For a finite ring R with identity and a finite unital R-module V the set C ( R ; V ) = { f : V → V | f ( α v ) = α f ( v ) C(R;V) = \{ f:V \to V|f(\alpha v) = \alpha f(v) for all α ∈ R , v ∈ V } \alpha \in R,v \in V\} is the centralizer near-ring determined by R and V. Those rings R such that C ( R ; V ) C(R;V) is a ring for every R-module V are characterized. Conditions are given under which C ( R ; V ) C(R;V) is a semisimple ring. It is shown that if C ( R ; V ) C(R;V) is a semisimple ring then C ( R ; V ) = End R ( V ) C(R;V) = {\text {End}_R}(V) .