On a conjecture by Herstein
type: Item
instance of: scholarly article
title: On a conjecture by Herstein
zbMATH Open document ID: 0688.16036
DOI: 10.1016/0021-8693(89)90322-0
author: Jer-Shyong Lin
author: Chen-Lian Chuang
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let P be the set of positive integers. Let R be an arbitrary ring, with N(R), \(N_ r(R)\), and Z(R) denoting respectively the nil radical, the sum of all nil right ideals, and the center. For elements \(x_ 1,x_ 2,..\). of R define extended commutators \([x_ 1,x_ 2,...,x_{k+1}]\) by \([x_ 1,x_ 2]=x_ 1x_ 2-x_ 2x_ 1\) and \([x_ 1,...,x_{n+1}]=[[x_ 1,...,x_ n],x_{n+1}]\); and if \(x_ 1=x\) and \(x_ 2=x_ 3=...=x_{k+1}=y\), abbreviate the notation to \([x,y]_ k\). For \(k\geq 1\), define \(T_ k(R)\) as the set of \(x\in R\) such that for each \(y\in R\), there exists \(n=n(x,y)\in P\) for which \([x,y^ n]_ k=0\); call R a \(C_ k\)-ring if for each \(x,y\in R\), P contains \(m=m(x,y)\) and \(n=n(x,y)\) for which \([x^ m,y^ n]_ k=0\). Extend these definitions by letting k vary with x and y, thereby obtaining the set T(R) and the notion of C-ring. The hypercenter theorem of \textit{I. N. Herstein} [J. Algebra 36, 151-157 (1975; Zbl 0313.16036)] asserts that if \(N(R)=\{0\}\), then \(T_ 1(R)=Z(R)\); and a subsequent result of Herstein [J. Algebra 38, 112-118 (1976; Zbl 0323.16014)] is that every \(C_ 1\)-ring with \(N(R)=\{0\}\) is commutative. It has long been conjectured that the same conclusions hold for \(T_ k(R)\) and \(C_ k\)-rings, and various special cases have been established. The present paper proves the conjectures in general - specifically, that (1) for each \(k\in P\), if \(N(R)=\{0\}\), then \(T_ k(R)=Z(R)\); and (2) for arbitrary \(k\in P\), every \(C_ k\)-ring with \(N(R)=\{0\}\) is commutative. Also included are proofs of the following: (3) if \(N_ r(R)=\{0\}\), then \(T(R)=Z(R)\); (4) if R is a C-ring with \(N_ r(R)=\{0\}\), then R is commutative. It is natural to conjecture that every C-ring with \(N(R)=\{0\}\) is commutative, and the authors conclude with a proof of equivalence of this conjecture with two other conjectures. This is an important paper. It is carefully argued and, given the difficulty of the problem, surprisingly easy to read.
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 4127453
zbMATH Keywords: commutativity
zbMATH Keywords: nil radical
zbMATH Keywords: nil right ideals
zbMATH Keywords: center
zbMATH Keywords: extended commutators
zbMATH Keywords: \(C_ k\)-ring
zbMATH Keywords: C-ring
zbMATH Keywords: hypercenter theorem
Wikidata QID: Q122923471
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90322-0
OpenAlex ID: W2022148550
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description: scientific article; zbMATH DE number 4127453