Mal'cev nilpotent algebras
type: Item
instance of: scholarly article
title: Mal'cev nilpotent algebras
zbMATH Open document ID: 0935.16011
author: David Riley
author: Vladimir Tasić
published in: Archiv der Mathematik
publication date: 2000-02-15T00:00:00Z
review text: The algebras under study are torsion free associative algebras over a commutative domain of prime characteristic \(p\) or of characteristic \(0\). Mal'cev introduced semigroup identities that hold in groups iff the groups are nilpotent. Given \(x\), \(y\), \(w_n\), \(n\geq 2\), define \(x_1=xy\), \(y_1=yx\), \(x_{n+1}=x_nw_{n+1}y_n\), \(y_{n+1}=y_nw_{n+1}x_n\). A semigroup is called Mal'cev nilpotent of class \(n\) if it satisfies the identity \(x_n=y_n\) and does not satisfy \(x_{n-1}=y_{n-1}\). If \(A\) is a ring, the circle semigroup of \(A\) is denoted by \((A;\circ)\) where \(x\circ y=x+y+xy\). Theorem 1: An algebra \(A\) is Lie nilpotent of class \(n\) iff \((A;\circ)\) is Mal'cev nilpotent of class \(n\). Theorem 2: If \(R\) is a ring that is Lie nilpotent of class \(n\), then the multiplicative semigroup and the circle semigroup of \(R\) are Mal'cev nilpotent of class at most \(n\). This generalizes a result of \textit{N. D. Gupta} and \textit{F. Levin} [J. Algebra 81, 225-231 (1983; Zbl 0514.16024)]. Theorem 3: Let \(A\) be an algebra with \(1\) that is generated by its nilpotent elements. Then its group of units is nilpotent of class at most \(n\) iff \(A\) is Lie nilpotent of class at most \(n\).
reviewed by: Awad A. Iskander
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 1292443
zbMATH Keywords: semigroup identities
zbMATH Keywords: Maltsev nilpotent semigroups
zbMATH Keywords: circle groups of rings
zbMATH Keywords: Lie nilpotent algebras
zbMATH Keywords: Maltsev nilpotent algebras
zbMATH Keywords: multiplicative semigroups of rings
zbMATH Keywords: nilpotent elements
zbMATH Keywords: groups of units
MaRDI profile type: Publication
full work available at URL: s000130050298
OpenAlex ID: W1983448047
DOI: 10.1007/S000130050298
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description: scientific article; zbMATH DE number 1292443