Modules over group rings of groups with restrictions on the system of all proper subgroups

Abstract

We consider the class $mathfrak M$ of $bf R$--modules where $bf R$ is an associative ring. Let $A$ be a module over a group ring $bf R$$G$, $G$ be a group and let $mathfrak L(G)$ be the set of all proper subgroups of $G$. We suppose that if $H in mathfrak L(G)$ then $A/C_{A}(H)$ belongs to $mathfrak M$. We investigate an $bf R$$G$--module $A$ such that $G not = G'$, $C_{G}(A) = 1$. We study the cases: 1) $mathfrak M$ is the class of all artinian $bf R$--modules, $bf R$ is either the ring of integers or the ring of $p$--adic integers; 2) $mathfrak M$ is the class of all finite $bf R$--modules, $bf R$ is an associative ring; 3) $mathfrak M$ is the class of all finite $bf R$--modules, $bf R$$=F$ is a finite field.