On the semigroup nilpotency and the Lie nilpotency of associative algebras
type: Item
instance of: scholarly article
title: On the semigroup nilpotency and the Lie nilpotency of associative algebras
zbMATH Open document ID: 0919.16026
DOI: 10.1007/BF02358975
published in: Mathematical Notes
publication date: 1999-09-08T00:00:00Z
review text: This paper studies the relation between Lie nilpotence and semigroup nilpotence of associative rings. An associative ring \(R\) can be viewed as a Lie ring \(R^{(-)}\) using the multiplication \([a,b]=ab-ba\), and so the notion of the Lie nilpotence of \(R^{(-)}\) makes sense. There are two associative semigroups related to \(R\): \(M(R)\) with multiplication that of \(R\) and \(A(R)\) with multiplication \(a\cdot b=a+b+ab\). There are various notions of nilpotence for a semigroup and the author uses the following, and variants of it, due to Mal'tsev. Using noncommuting indeterminates set \(U_0=x\), \(V_0=y\), and for \(k\geq 0\), \(U_{k+1}=U_kz_kV_k\) and \(V_{k+1}=V_kz_kU_k\). Then a semigroup is nilpotent of length \(m\) if \(U_{m+1}=V_{m+1}\) is an identity but \(U_m=V_m\) is not. The main results show that: whenever \(R^{(-)}\) is nilpotent of length \(m\) then \(M(R)\) and \(A(R)\) are nilpotent of length at most \(m\); if \(R\) is an algebra over an infinite field then the nilpotence of \(R^{(-)}\) of length \(m\) and the nilpotence of \(A(R)\) of length \(m\) are equivalent; and for any \(R\), \(A(R)\) nilpotent of length 2 implies that \(R^{(-)}\) is also. In the second statement \(M(R)\) may replace \(A(R)\) when \(1\in R\) since then \(A(R)\cong M(R)\). An example is given of an algebra \(R\) without identity but over an infinite field with \(M(R)\) nilpotent of length 2 but \(R^{(-)}\) not nilpotent.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M11
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 1257136
zbMATH Keywords: Lie nilpotence
zbMATH Keywords: semigroup nilpotence
zbMATH Keywords: associative rings
zbMATH Keywords: Lie rings
author: Alexei Krasilnikov
MaRDI profile type: Publication
cites work: Subsemigroups of nilpotent groups
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cites work: Q5850574
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full work available at URL: https://doi.org/10.1007/bf02358975
OpenAlex ID: W1990806244
description: scientific article; zbMATH DE number 1257136