Semilocal rings with \(n\)-Engel multiplicative group.
type: Item
instance of: scholarly article
title: Semilocal rings with \(n\)-Engel multiplicative group.
zbMATH Open document ID: 1070.16022
author: Bernhard Amberg
author: Y. P. Sysak
published in: Archiv der Mathematik
publication date: 2005-02-25T00:00:00Z
review text: Let \(R\) be an associative ring with a unit element \(1\). \(R\) is called local if the factor ring \(R/J\) is a division ring and \(R\) is semilocal if \(R/J\) is an Artinian ring. Every ring \(R\) may be considered as a Lie ring under the Lie multiplication \([r_1,r_2]=r_1r_2-r_2r_1\) for all \(r_1,r_2\in R\). If \(r_1,r_2,\dots\) are elements of \(R\), the Lie-commutators \([r_1,r_2,\dots,r_{n+1}]\) are defined inductively by \([r_1,r_2,\dots,r_{n+1}]=[[r_1,r_2,\dots,r_n],r_{n+1}]\) for all \(n\geq 2\). The ring \(R\) is called Lie nilpotent of class \(n\), if \([r_1,r_2,\dots,r_{n+1}]=0\) for all elements \(r_1,r_2,\dots,r_{n+1}\) of \(R\) and \(n\) is the least integer with this property. Moreover, \(R\) is Engel if \([r,s,\dots,s]=0\) for each pair of elements \(r\) and \(s\) of \(R\), and \(n\)-Engel if \(s\) appears exactly \(n\)-times. Also, \(R\) is a locally Lie nilpotent ring if every finitely generated subring of \(R\) is Lie nilpotent. Nilpotent, Engel and \(n\)-Engel groups are defined in a similar way where the usual group commutator replaces the Lie-commutator. In this paper the authors prove two major results: Theorem A. The multiplicative group \(R^*\) of a local ring \(R\) is \(m\)-Engel for some positive integer \(m\) if and only if \(R\) is \(n\)-Engel as a Lie ring for some positive integer \(n\) depending only on \(m\). Theorem B. Let \(R\) be a ring with \(1\) whose multiplicative group \(R^*\) is \(m\)-Engel for some integer \(m\geq 1\) and let \(J\) be the Jacobson radical of \(R\). If the factor ring \(R/J\) is commutative and \(R\) is generated by \(R^*\) as a ring, then \(R\) is an \(n\)-Engel ring for some integer \(n\geq 1\) depending only on \(m\). As a consequence of Theorem B the authors obtain the following Corollary. Let \(R\) be a semilocal ring whose multiplicative group \(R^*\) is \(m\)-Engel for some integer \(m\geq 1\). Then \(R^*\) is a locally nilpotent group and, furthermore, \(R\) is \(n\)-Engel as a Lie ring, provided …
reviewed by: Halina France-Jackson
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 2138529
zbMATH Keywords: local rings
zbMATH Keywords: semilocal rings
zbMATH Keywords: locally Lie nilpotent rings
zbMATH Keywords: Engel rings
zbMATH Keywords: \(n\)-Engel rings
zbMATH Keywords: Engel groups
zbMATH Keywords: \(n\)-Engel groups
zbMATH Keywords: multiplicative groups of rings
zbMATH Keywords: Jacobson radical
zbMATH Keywords: Levitzki radical
zbMATH Keywords: Lie nilpotent ideals
MaRDI profile type: Publication
full work available at URL: s00013-004-1113-0
OpenAlex ID: W2079621399
DOI: 10.1007/S00013-004-1113-0
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description: scientific article; zbMATH DE number 2138529