On one class of modules that are close to Noetherian

We consider an $\mathbf RG$-module $A$ over a commutative Noetherian ring $\mathbf R$. Let $G$ be a group having infinite section $p$-rank (or infinite 0-rank) such that $C_G(A)=1$, $A/C_A(G)$ is not a Noetherian $\mathbf R$-module, but the quotient $A/C_A(H)$ is a Noetherian $\mathbf R$-module for every proper subgroup $H$ of infinite section $p$-rank (or infinite 0-rank, respectively). In this paper, it is proved that if $G$ is a locally soluble group, then $G$ is soluble. Some properties of soluble groups of this type are also obtained.