On one class of modules over group rings with finiteness restrictions

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.