Radical rings with Engel conditions
type: Item
instance of: scholarly article
title: Radical rings with Engel conditions
zbMATH Open document ID: 0969.16003
author: Bernhard Amberg
author: Y. P. Sysak
published in: Journal of Algebra
publication date: 2001-02-28T00:00:00Z
review text: In a ring \(R\) the Lie multiplication and the Lie commutators are defined inductively by \([r,s]=rs-sr\), and \([r_1,\dots,r_{n+1}]=[[r_1,\dots,r_n],r_{n+1}]\). A ring \(R\) is \(n\)-Engel, if \([r,s,\dots,s]=0\) where \(s\) appears exactly \(n\) times. \(n\)-Engel groups are defined correspondingly where the usual group commutator replaces the Lie commutator. It is proved that for a Jacobson radical ring \(R\) the adjoint group \(R^*\) is \(n\)-Engel if and only if \(R\) is \(m\)-Engel where \(m\) depends on \(n\). The proof is achieved by several Lemmas, some of technical character, others of interest on their own right (e.g. the factor ring \(R/L\) of an \(n\)-Engel ring \(R\) by its Levitzki radical \(L\) is commutative).
reviewed by: R. Wiegandt
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 1525277
zbMATH Keywords: Engel conditions
zbMATH Keywords: Lie multiplication
zbMATH Keywords: Lie commutators
zbMATH Keywords: \(n\)-Engel groups
zbMATH Keywords: Jacobson radical rings
zbMATH Keywords: adjoint groups
zbMATH Keywords: \(n\)-Engel rings
zbMATH Keywords: Levitzki radical
MaRDI profile type: Publication
OpenAlex ID: W2046482056
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DOI: 10.1006/JABR.2000.8370
description: scientific article; zbMATH DE number 1525277