On some modules over group rings of locally soluble groups with rank restrictions on subgroups

The author studies an $bf R$$G$-module $A$ such that $bf R$ is an integral domain, $G$ is a locally soluble group of infinite section $p$-rank (or infinite 0-rank), $C_{G}(A)=1$, $A/C_{A}(G)$ is not a noetherian $bf R$-module, 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 noetherian $bf R$-module. It is proved that under the above conditions, $G$ is a soluble group. Some properties of soluble groups of this type are obtained.