Modules over integer group rings of soluble groups that are close to minimax modules
O. Dashkova
Investigation of modules over group rings is an important direction of modern algebra. Artinian and Noetherian modules over group rings are a broad class of modules over group rings. The class of minimax modules [1, chap. 7] is a generalization of Artinian and Noetherian modules. Let \(A\) be an \(\bf R\)-module, where \(\bf R\) is a ring, and let \(G\) be a group. \(A\) is called a minimax \(\bf R\)-module if \(A\) has a series of submodules with every factor either an Artinian or a Noetherian
\(\bf R\)-module. Thus arises the problem of investigating the modules over group rings which are not minimax but are similar to minimax modules in some sense.
The main result of the work is the following theorem.
Theorem.Let \(A\) be a \(\mathbb{Z}G\)-module, where \(\mathbb{Z}\) is the ring of integers, \(G\) is an infinite soluble group, and \(C_{G}(A) =1\). If \(A/C_{A}(G)\) is not a minimax \(\mathbb{Z}\)-module and, for every proper subgroup \(H\) of \(G\), the quotient module \(A/C_{A}(H)\) is a minimax \(\mathbb{Z}\)-module then \(G\) is isomorphic to \(C_{q^{\infty}}\) for some prime \(q\).
We construct an example of a \(\mathbb{Z}G\)-module \(A\) with the mentioned properties where \(G\) is isomorphic to \(C_{q^{\infty}}\) for some prime \(q\).
References
L. A. Kurdachenko, I. Ya. Subbotin, N. N. Semko, Insight
into Modules over Dedekind Domains. Kyev: National Academy
of Sciences of Ukraine, Institute of Mathematics (2008).
