On some center-like subsets of rings
type: Item
instance of: scholarly article
title: On some center-like subsets of rings
zbMATH Open document ID: 0622.16014
DOI: 10.1007/BF01189630
author: Howard E. Bell
author: Itzhak Nada
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: The authors obtain some interesting commutativity results involving higher commutators. Specifically, for any ring R and any x,y\(\in R\), define \([x,y]_ 0=x\), \([x,y]_ 1=xy-yx\), and for \(k>1\), \([x,y]_ k=[[x,y]_{k-1},y]\). Set \(T(R)=\{x\in R|\) for each \(y\in R\) there are n and k depending on both x and y so that \([x,y^ n]_ k=0\}\) and \(K(R)=\{x\in R|\) for some n and k depending on x, \([x,y^ n]_ k=0\) for all \(y\in R\}\). Finally, for each positive integer k let \(B_ k(R)=\{x\in T(R)|\) for each \(y\in R\) \([x,y^ n]_ k=0\}\). Clearly, each of these sets contains Z(R), the center of R, and the main results give conditions which force each to be equal to Z(R). In particular, \(T(R)=Z(R)\) when R is a semi-simple ring; \(K(R)=Z(R)\) when R is a semi- prime ring; and \(B_ k(R)=Z(R)\) when it is a subring of R, provided that R has an identity element and contains no nonzero nil ideals.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 4008602
zbMATH Keywords: commutativity
zbMATH Keywords: higher commutators
zbMATH Keywords: center
zbMATH Keywords: semi-simple ring
zbMATH Keywords: semi-prime ring
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01189630
OpenAlex ID: W2035690409
description: scientific article; zbMATH DE number 4008602