Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings. I
type: Item
instance of: scholarly article
title: Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings. I
zbMATH Open document ID: 0536.16023
DOI: 10.1016/0021-8693(84)90087-5
author: A. I. Lichtman
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: The goal here is to show that the ring \(D_ n\) of \(n\times n\) matrices over the division ring of fractions D of a universal enveloping algebra enjoys properties similar to that of a PI-algebra over a field. Let H be a Lie algebra over the field K, let U(H) be its universal enveloping algebra and let D be the ring of fractions of U(H). The main result is as follows. Theorem A. Suppose that H satisfies one of the three conditions: i. U(H) is an Ore ring and \(\cap^{\infty}_{m=1}H^ m=0\). ii. H is solvable of class 2. iii. H is finite dimensional over K. Then \(D_ n\) satisfies: 1. The Jacobson radical of any finitely generated subring of \(D_ n\) is nilpotent of index \(\leq n.\) 2. If V is any finitely generated subring of \(D_ n\) such that any element of V is a sum of nilpotent elements of V, then \(V^ n=0.\) 3. Any noncentral normal subgroup of \(GL_ n(D)\) contains a noncyclic free subgroup. 4. Any periodic subgroup of \(GL_ n(D)\) is locally finite. 5.If \(char K=0\) then any periodic subgroup of \(GL_ n(D)\) contains an Abelian normal subgroup of index \(\leq \rho(n)\) where \(\rho\) is the Jordan function. It is also shown that if H is solvable with \(\cap^{\infty}_{n=1}H^ n=0\), then any nilpotent subgroup of \(D^*\) is Abelian. Moreover, when \(char K=0,\) then any nilpotent-by-finite subgroup of \(D^*\) is Abelian. Finally the paper considers division rings of fractions of group rings. Let G be a solvable residually torsion free nilpotent group or a torsion free polycyclic-by-finite group. If D is the division ring of fractions of KG, then \(D_ n\) also satisfies the conclusions of Theorem A.
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3851318
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: PI-algebra
zbMATH Keywords: Lie algebra
zbMATH Keywords: Ore 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: Jordan function
zbMATH Keywords: nilpotent-by-finite subgroup
zbMATH Keywords: division rings of fractions of group rings
zbMATH Keywords: solvable residually torsion free nilpotent group
zbMATH Keywords: torsion free polycyclic-by-finite group
reviewed by: D. S. Passman
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90087-5
OpenAlex ID: W2015767405
description: scientific article; zbMATH DE number 3851318