On modules over group rings of locally soluble groups for a ring of \(p\)-adic integers

Abstract

The author studies the \(\bf Z_{p^{\infty}}\)\(G\)-module \(A\) such that \(\bf Z_{p^{\infty}}\) is a ring of \(p\)-adic integers, a group \(G\) is locally soluble, the quotient module \(A/C_{A}(G)\) is not Artinian \(\bf Z_{p^{\infty}}\)-module, and the system of all subgroups \(H \leq G\) for which the quotient modules \(A/C_{A}(H)\) are not Artinian \(\bf Z_{p^{\infty}}\)-modules satisfies the minimal condition on subgroups.  It is proved that the group \(G\) under consideration is soluble and some its properties are obtained.