A note on \(k\)-th commutators in an associative ring
type: Item
instance of: scholarly article
title: A note on \(k\)-th commutators in an associative ring
zbMATH Open document ID: 0903.16017
DOI: 10.1007/BF02844726
publication date: 1999-01-05T00:00:00Z
review text: Set \([x,y]=[x,y]_1=xy-yx\), and for \(k>1\), \([x,y]_{k+1}=[[x,y]_k,y]\). The main theorem in this paper shows that if \(k>0\) is fixed, \(R\) is a 2-torsion free ring containing no nonzero nil right ideal, and for all \(a,b,c,d\in R\), \(f(a,b,c,d)=[[a,b],[c,d]]_k\) is nilpotent, then \(R\) is commutative. Consequences of this result are that if \(f(a,b,c,d)^n\) is in the center of \(R\) with \(n=n(a,b,c,d)\), then \(R\) embeds in \(M_2(C)\) for \(C\) a commutative ring, and if instead \(f(a,b,c,d)\) is either nilpotent or regular then \(R\) is a domain, is commutative, or is an order in \(M_2(F)\) for \(F\) a field. A corollary extends the main theorem to the case when \(a,b,c,d\in U\), a noncentral Lie ideal of \(R\).
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 1172077
zbMATH Keywords: commutativity theorems
zbMATH Keywords: 2-torsion free rings
zbMATH Keywords: center
zbMATH Keywords: nilpotent elements
zbMATH Keywords: regular elements
zbMATH Keywords: noncentral Lie ideals
reviewed by: Charles Lanski
MaRDI profile type: Publication
cites work: Rings with nil and power central \(k\)-th commutators
cites work: On \(n\)-th commutators with nilpotent or regular values in rings
cites work: On the Lie structure of an associative ring
cites work: Q4112071
cites work: PI-algebras. An introduction
cites work: Q4025352
cites work: Q5677619
cites work: Q3910639
published in: Rendiconti del Circolo Matematico di Palermo
author: M. Onofrio Di Vincenzo
description: scientific article; zbMATH DE number 1172077