On a class of modules over group rings of locally soluble groups

A module $A$ over a group ring $DG$ is studied in the case when $D$ is a Dedekind domain, the group $G$ is locally soluble, the quotient module $A/C_A(G)$ is not an Artinian $D$-module, and the system of all subgroups $H\le G$ for which the quotient modules $A/C_A(H)$ are not Artinian $D$-modules satisfies the minimality condition for subgroups. Under these assumptions, it is proved that the group $G$ is hyperabelian and some properties of its periodic part are described.