On matrix rings and linear groups over fields of fractions of group rings and enveloping algebras. II
type: Item
instance of: scholarly article
title: On matrix rings and linear groups over fields of fractions of group rings and enveloping algebras. II
zbMATH Open document ID: 0544.16010
DOI: 10.1016/0021-8693(84)90187-X
author: A. I. Lichtman
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 3865510
zbMATH Keywords: solvable residually torsion free nilpotent group
zbMATH Keywords: group ring
zbMATH Keywords: Ore domain
zbMATH Keywords: division ring of fractions
zbMATH Keywords: matrix ring
zbMATH Keywords: Jacobson radical
zbMATH Keywords: finitely generated subring
zbMATH Keywords: sum of nilpotent elements
zbMATH Keywords: noncentral normal subgroup
zbMATH Keywords: noncyclic free subgroup
zbMATH Keywords: periodic subgroup
zbMATH Keywords: Jordan function
zbMATH Keywords: solvable-by- finite torsion free group
reviewed by: D. S. Passman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3865510