Radical rings with soluble adjoint groups
type: Item
instance of: scholarly article
title: Radical rings with soluble adjoint groups
zbMATH Open document ID: 1010.16018
author: Y. P. Sysak
author: Bernhard Amberg
published in: Journal of Algebra
publication date: 2002-06-30T00:00:00Z
review text: An associative ring \(R\) is called radical if \(R\) coincides with its Jacobson radical, that is, when the set of all elements of \(R\) forms a group (denoted by \(R^\circ\) and called the adjoint group of \(R\)) under the operation \(r\circ s=r+s+rs\) on \(R\). 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 \([r_1,r_2]=r_1r_2-r_2r_1\) and \([r_1,r_2,\dots,r_{n+1}]=[[r_1,r_2,\dots,r_n],r_{n+1}]\) for all \(n\geqslant 2\). Every associative ring \(R\) can be considered as a Lie ring under the Lie multiplication \([r_1,r_2]=r_1r_2-r_2r_1\) for all \(r_1,r_2\in R\). For any additive subgroups \(V\) and \(W\) of \(R\), the additive subgroup of \(R\) generated by all Lie commutators \([r_1,r_2]\) with \(r_1\in V\) and \(r_2\in W\) is denoted by \([V,W]\). The derived chain of a Lie ring \(R\) is defined inductively by \(\delta_0(R)=R\) and \(\delta_{n+1}(R)=[\delta_n(R),\delta_n(R)]\) for each integer \(n\geqslant 0\). The ring \(R\) is called Lie-soluble of length at most \(m\) if \(\delta_m(R)=0\). Soluble groups of derived length \(m\) are defined in a corresponding way where the usual group commutator replaces the Lie commutator. A ring \(R\) is called an \(n\)-Engel ring if \([r,s,\dots,s]=0\) for each pair of elements \(r\) and \(s\) of \(R\) where \(s\) appears exactly \(n\)-times. \(m\)-Engel groups are defined correspondingly. An ideal \(I\) of \(R\) is called a commutative ideal if \([I,I]=0\). The authors prove that for a radical ring \(R\), the adjoint group \(R^\circ\) is soluble of derived length at most \(n\) for some positive integer \(n\) if and only if the following two conditions hold: (1) \(R\) is Lie-soluble of length at most \(l\) for some positive integer \(l\) depending only on \(n\), and (2) there exists a chain \(0=I_0\subseteq I_1\subseteq\cdots\subseteq I_m=R\) of ideals of \(R\) such that every factor \(I_i/I_{i-1}\) is generated by commutative ideals of \(R/I_{i-1}\) fo…
reviewed by: Halina France-Jackson
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F45
zbMATH DE Number: 1740521
zbMATH Keywords: radical rings
zbMATH Keywords: adjoint groups
zbMATH Keywords: Lie-soluble rings
zbMATH Keywords: soluble groups
zbMATH Keywords: Lie commutators
zbMATH Keywords: Engel rings
zbMATH Keywords: Engel groups
zbMATH Keywords: locally nilpotent rings
MaRDI profile type: Publication
full work available at URL: jabr.2001.8996
OpenAlex ID: W2031493129
cites work: Subgroups of the Adjoint Group of a Radical Ring
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cites work: Complex analysis and convolution operators
cites work: Lie Solvable Rings
cites work: Lie centrally metabelian group rings
cites work: Q3964737
DOI: 10.1006/JABR.2001.8996
description: scientific article; zbMATH DE number 1740521