On almost-quasi-commutative rings.
type: Item
instance of: scholarly article
title: On almost-quasi-commutative rings.
zbMATH Open document ID: 1181.16033
published in: Acta Mathematica Hungarica
publication date: 2009-12-28T00:00:00Z
review text: Let \(R\) be an associative ring and for any elements \(x,y\in R\) there exist a nonnegative integer \(n=n(x,y)\) and nonzero relatively prime integers \(j(x,y)\) and \(k(x,y)\) such that \(jxy=k(yx)^n\). The authors prove the following main results: 1) the commutator ideal of \(R\) is nil, any idempotent of \(R\) is central; 2) if \(R\) is torsion-free and for any \(x,y\in R\) \(n(x,y)\geq 2\) then \(R\) is commutative; 3) if \(R\) is torsion-free and for any \(x,y\in R\) \(n(x,y)=1\) then \(R\) is either commutative or nil; 4) if the set of all nilpotent elements of a periodic ring \(R\) is commutative and \(n=n(x,y)\geq 1\) is a fixed number for any \(x,y\in R\) then \(R\) is commutative; 5) if \(R\) is a ring with \(1\) and for any \(x,y\in R\) \(n=n(x,y)\) is a fixed integer and either \(n\leq 2\) or \(n\geq 3\) and \(R\) is \((n-1)\)-torsion-free then \(R\) is commutative.
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5651861
zbMATH Keywords: almost-quasi-commutative rings
zbMATH Keywords: commutator ideal
zbMATH Keywords: periodic rings
zbMATH Keywords: generalized quasi-periodic rings
zbMATH Keywords: \(J\)-rings
zbMATH Keywords: nilpotent elements
zbMATH Keywords: commutativity theorems
author: Howard E. Bell
author: A. Klein
reviewed by: Yu. N. Maltsev
MaRDI profile type: Publication
DOI: 10.1007/S10474-009-7238-Z
description: scientific article; zbMATH DE number 5651861