Just infinite modules over locally soluble groups (Q1113283)

A module over a ring is called just infinite if it is infinite, all proper quotients are finite, and the intersection of all the non-zero submodules is zero. (The third condition is inserted to exclude infinite simple modules.) In many group theoretic investigations one has to consider a group G with an abelian normal subgroup A such that A is just infinite as a \({\mathbb{Z}}(G/A)\)-module. The authors are concerned with just infinite modules over group rings of locally soluble groups. The basic result is Theorem 1. Let J be a principal ideal domain which is just infinite as a J-module. Let G be a locally soluble group and A a JG- module which is J-torsion-free. Then there are embeddings \(J\hookrightarrow \hat J\), \(A\hookrightarrow \hat A\) where \(\hat J\) is an integral domain and \(\hat A\) is a free \(\hat J-\)module of finite rank such that if A is a just infinite JG-module, then \(\hat A\) is just infinite as a ĴG-module. An easy corollary is that \(G/C_ G(A)\) is abelian-by-finite whenever A is a just infinite JG-module. In fact \(\hat J\) and \(\hat A\) are the inverse limits \(\lim_{\leftarrow}(J/Jf^ k)\), \(\lim_{\leftarrow}(A/Af^ k)\) where f is a prime of J such that \(A\neq Af.\) Some applications are given, for example, Theorem 2. Let G be a locally soluble group with finite torsion-free rank, and let A be a just infinite \({\mathbb{Z}}G\)-module which is \({\mathbb{Z}}\)-torsion-free. Then \(G/C_ G(A)\) is abelian-by-finite and \(A^+\) is a minimax group. - Theorem 3. Let G be a torsion-free, hyperabelian, residually finite group. If G satisfies min- \(\infty\) for normal subgroups, (i.e. in any descending chain of normal subgroups almost all the factors are finite), then G is a minimax group.