Some properties on rings with units satisfying a group identity
type: Item
instance of: scholarly article
title: Some properties on rings with units satisfying a group identity
zbMATH Open document ID: 0967.16012
author: Chia-Hsin Liu
published in: Journal of Algebra
publication date: 2001-07-16T00:00:00Z
review text: A ring \(R\) with identity element 1 and group of units \(U(R)\) satisfies a group identity, or is a GI-ring, if for some nontrivial word \(\omega(x_1,\dots,x_n)\) in the free group \(\langle x_1,\dots,x_n,\dots\rangle\) and all \(u_j\in U(R)\), \(\omega(u_1,\dots,u_n)=1\). The author proves that GI-rings satisfy some of the same properties as rings satisfying a polynomial identity (PI-rings). The main results assume that \(R\) is a GI-ring and show: if \(a,b\in R\) with \(ab=1\) then \(ba=1\); if \(R\) is a semiprime algebra over an infinite field then \(R\) has bounded index of nilpotence; and if \(R\) is an algebra which is algebraic over the field \(K\) then \(R\) is locally finite. In the last case, if \(K\) is infinite or if the sum \(N(R)\) of all nilpotent ideals of \(R\) is nilpotent then \(R\) is a PI-ring.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 1539384
zbMATH Keywords: PI-rings
zbMATH Keywords: groups of units
zbMATH Keywords: group identities
zbMATH Keywords: GI-rings
zbMATH Keywords: semiprime algebras
zbMATH Keywords: nilpotent ideals
MaRDI profile type: Publication
full work available at URL: jabr.2000.8397
OpenAlex ID: W2081233839
cites work: Rational identities and applications to algebra and geometry
cites work: Nonmatrix varieties and nil-generated algebras whose units satisfy a group identity
cites work: Group identities on units of rings
cites work: Group algebras whose units satisfy a group identity
cites work: Symmetric units and group identities
cites work: Q3999650
cites work: Conjugates in Prime Rings
cites work: Group identities on units of locally finite algebras and twisted group algebras
cites work: Group algebras with units satisfying a group identity
cites work: Group algebras with units satisfying a group identity II
cites work: Group algebras whose units satisfy a group identity. II
cites work: Q3961021
cites work: On free subgroups of units of rings
DOI: 10.1006/JABR.2000.8397
description: scientific article; zbMATH DE number 1539384