The decomposition of minimax modules over hyperfinite groups (Q1312363)

Let \(G\) be a locally soluble hyperfinite group. The \({\mathbf Z} G\)-module \(A\) is minimax if it has a finite series of \({\mathbf Z} G\)-submodules \(0 = A_ 0 \subseteq A_ 1 \subseteq \cdots \subseteq A_ n = A\) such that each factor \(F_ i = A_ i / A_{i - 1}\) is either an artinian or a noetherian \({\mathbf Z} G\)-module. It is shown that \(A\) has an \(f\)- decomposition, i.e. \(A = A^ f \oplus A^{\overline{f}}\), where \(A^ f\) is an \({\mathbf Z}G\)-submodule of \(A\) such that every irreducible \({\mathbf Z}G\)-factor of \(A^ f\) is finite and the \({\mathbf Z} G\)-module \(A^{\overline{f}}\) has no nonzero finite \({\mathbf Z} G\)-factors. By results of Zaitsev and the first author it is known that each factor \(F\) of \(A\) has such an \(f\)-decomposition. To extend this to \(A\) one has to consider modules which are either noetherian-by-artinian or artinian-by- noetherian.