On matrix rings and linear groups over fields of fractions of group rings and enveloping algebras. II
This article continues the author's work on linear groups over certain division rings [Part I, cf. ibid. 88, 1-37 (1984; Zbl 0536.16023)]. Let H be a solvable residually torsion free nilpotent group. Then the group ring K[H] is an Ore domain with division ring of fractions D. The following properties of the matrix ring \(D_ n\) are established: (1) If S is a finitely generated subring of \(D_ n\), then the Jacobson radical of S is nilpotent of index \(\leq n\). (2) If T is a finitely generated subring of \(D_ n\) such that any element of T is a sum of nilpotent elements, then \(T^ n=0\). (3) Any noncentral normal subgroup of \(GL_ n(D)\) contains a noncyclic free subgroup. (4) Any periodic subgroup \(G\subseteq GL_ n(D)\) is locally finite. (5) If char K\(=0\), then any periodic subgroup \(G\subseteq GL_ n(D)\) contains an abelian normal subgroup of index \(\leq \rho(n)\) where \(\rho\) (n) is the Jordan function. Now let H be a finitely generated solvable-by-finite torsion free group and assume that K[H] is an Ore domain with division ring of fractions D. If H is not nilpotent-by-finite, then it is shown that D contains K[F] the group ring of a free group F of countable rank. As a consequence of this one concludes that matrix rings over K[F] satisfy properties (1), (2), (4) and (5) listed above.
solvable residually torsion free nilpotent group
scientific article; zbMATH DE number 3865510
scientific article