On associative algebras satisfying the Engel condition
type: Item
instance of: scholarly article
title: On associative algebras satisfying the Engel condition
zbMATH Open document ID: 0701.16026
DOI: 10.1007/BF02764947
author: Aner Shalev
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: \textit{A. I. Kostrikin} [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 3-34 (1959; Zbl 0090.245)] proved that an n-Engel Lie algebra of characteristic p is locally nilpotent if \(p>n\) or \(p=0\). \textit{E. Zel'manov} [Sib. Mat. Zh. 29, No.5, 112-117 (1988; Zbl 0673.17006)] proved that for \(p=0\) such an algebra is even nilpotent, while for \(p\neq 0\) \textit{Yu. Razmyslov}'s [Algebra Logika 10, 33-44 (1971; Zbl 0253.17005)] examples show that it may be nonsolvable. Recently Zel'manov proved that such an algebra is always locally nilpotent (with no restriction on p and n). Here the author proves that an n-Engel Lie algebra \(A^{(-)}\) arising from a finitely generated associative algebra A is Lie-nilpotent. This result is not covered by Zel'manov's one as \(A^{(-)}\) need not be finitely generated. It is also proved that the group of units U(A) is contained in some locally nilpotent variety. The proof is based on some results on PI- algebras such as \textit{A. Braun}'s [J. Algebra 89, 375-396 (1984; Zbl 0538.16013)] on the nilpotency of the radical and Shirshov's solution of Kurosh's problem.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 20F45
zbMATH DE Number: 4148352
zbMATH Keywords: n-Engel Lie algebra
zbMATH Keywords: locally nilpotent
zbMATH Keywords: finitely generated associative algebra
zbMATH Keywords: Lie-nilpotent
zbMATH Keywords: group of units
zbMATH Keywords: locally nilpotent variety
zbMATH Keywords: PI- algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4148352