On rings of quotients of group algebras of soluble groups of finite rank (Q1181502)

Let \(\mathfrak T\) be the class of all finite extensions of torsion-free soluble groups. Let \(\mathfrak X\) be the class of groups \(G\) which are residually (torsion free \(\mathfrak T\)) and which have a finite series \(\{1\}\leq \dots G_ i\leq \dots G_ n \leq G\) with \(G/G_ n\) a \(\mathfrak T\)-group and each \(G_{i+1}/G_ i\) residually (torsion-free abelian of finite rank). Let \(G\) be a group which is locally (a finite extension of an \(\mathfrak X\) group). The class of such groups \(G\) includes, for example, all soluble-by-finite linear groups over the rationals and all metabelian groups with torsion-free derived group. Let \(F\) be a field of characteristic \(p\) and suppose that \(G\) has no section which is a divisible torsion-free abelian \(p\)-group. The group algebra \(FG\) has a classical ring of quotients. It is shown that the matrix ring \(D^{n\times n}\) is locally residually finite dimensional over \(F\). This generalises Theorem 1 of the author's earlier work [in Arch. Math. 57, No. 3, 228-237 (1991; Zbl 0702.20005)] and earlier work of \textit{A. I. Lichtman} [Isr. J. Math. 42, 318-326 (1982; Zbl 0504.16013)]. The Theorem has a number of corollaries -- on matrix groups over \(D\), for example.