On locally soluble \(\mathrm A \mathrm F \mathrm N\)-groups

Abstract

Let \(A\) be an \({\mathbf{R}G}\)-module, where \(\bf R\) is a commutative ring, \(G\) is a locally soluble group, \(C_{G}(A)=1\), and each proper subgroup \(H\) of \(G\) for which \(A/C_{A}(H)\) is not a noetherian \(\bf R\)-module, is finitely generated. We describe the structure of a locally soluble group \(G\) with these conditions and the structure of \(G\) under consideration if \(G\) is a finitely generated soluble group and the quotient module \(A/C_{A}(G)\) is not a noetherian \(\bf R\)-module.