On one class of modules over group rings with finiteness restrictions

Abstract

The author studies the $bf R$$G$-module $A$ such that $bf R$ is an associative ring‎, ‎a group $G$ has infinite section $p$-rank (or infinite 0-rank)‎, ‎$C_{G}(A)=1$‎, ‎and for every‎ ‎proper subgroup $H$ of infinite section $p$-rank (or infinite 0-rank respectively) the quotient module $A/C_{A}(H)$ is‎ ‎a finite $bf R$-module‎. ‎It is proved that if the group $G$ under‎ ‎consideration is locally soluble‎ ‎then $G$ is a soluble group and $A/C_{A}(G)$ is a finite $bf R$-module‎. ‎