Modules over group rings of soluble groups with a certain condition of maximality. (Q657393)

Let \(A\) be an \(RG\)-module, suppose that \(R\) is an integral domain and \(G\) is a soluble group. Suppose that \(C_G(A)=1\) and \(A/C_A(G)\) is not a Noetherian \(R\)-module. Let \(L_{nnd}(G)\) be the family of all subgroups \(H\) of \(G\) such that \(A/C_A(H)\) is not a Noetherian \(R\)-module. This paper is concerned with the structure of groups \(G\) for which \(L_{nnd}(G)\) satisfies the maximal condition. The group \(G\) is then said to satisfy max-nnd. As an example of some of the results obtained there is: Let \(A\) be an \(RG\)-module and suppose that the group \(G\) satisfies the condition max-nnd. If \(G\neq G'\) then either \(G/G'\) is finitely generated or \(G/G'\) contains a finitely generated subgroup \(S/G'\) such that \(G/S\) is a quasicyclic \(p\)-group for some prime \(p\). The paper has some very nice results.