On rings with redundancy in multiplication
type: Item
instance of: scholarly article
title: On rings with redundancy in multiplication
zbMATH Open document ID: 0665.16021
DOI: 10.1007/BF01261970
author: Howard E. Bell
author: A. Klein
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: In this paper, the commutativity problem of rings is studied from the standpoint of a criterion of redundancy the introduction of which was stimulated by the work of \textit{G. A. Freiman} [Aequationes Math. 22, 140-152 (1981; Zbl 0489.20020)]. Let \(S\) be an arbitrary finite subset of the ring \(R\), \(S^ n=\{x_ 1\cdots x_ n|\) \(x_ i\in S\}\), \(n>1\), \(C(R)\) the commutator ideal of \(R\). The set \(S\) is called \(n\)-redundant if \(| S^ n| <| S|^ n\), and \(R\) is \((m,n)\)-redundant if every subset \(S\subset R\) with \(| S| =m\) is \(n\)-redundant. The main theorem asserts that any \((2,2)\)-redundant ring with a unit is commutative. The authors also show that \(C(R)\) is a nil-ideal if \(R\) is any \((m,n)\)-redundant algebra over an infinite field or an arbitrary \((2,3)\)- or \((3,2)\)-redundant ring. Proceeding from different assumptions the last result for rings was obtained by the authors and \textit{T. Jenkyns} and \textit{C. Laywine} [Result. Math. 13, 12-22 (1988; Zbl 0646.16026)].
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 4087870
zbMATH Keywords: \((m,n)\)-redundant rings
zbMATH Keywords: commutativity
zbMATH Keywords: redundancy
zbMATH Keywords: commutator ideals
zbMATH Keywords: nil-ideals
zbMATH Keywords: \((m,n)\)-redundant algebras
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
cites work: On some commutativity theorems of Herstein
cites work: Rings with permutable products
cites work: On two- and three-element subsets of groups
full work available at URL: https://doi.org/10.1007/bf01261970
OpenAlex ID: W2005186099
description: scientific article; zbMATH DE number 4087870