Nilpotency and generalized Lie ideals
type: Item
instance of: scholarly article
title: Nilpotency and generalized Lie ideals
zbMATH Open document ID: 0685.16020
DOI: 10.1016/0021-8693(89)90287-1
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/1828/1557
review text: The authors generalize the notion of inner Lie ideal, defined by \textit{G. M. Benkart} [J. Algebra 43, 561-584 (1976; Zbl 0342.16009)], to that of generalized Lie ideal (GLI), and prove a structure theorem for such objects. For a nonempty subset V of a ring R, set \(V^{(1)}(R)=[V,R]\), and in general, \(V^{(n+1)}(R)=[V,V^{(n)}(R)]\). Call V a GLI of index n if V is an additive subgroup of R satisfying \(V^{(n)}(R)\subset V.\) The main result on the structure of GLIs is: Theorem. Let R be a prime ring with char R\(=0\), Z the center of R, C the extended centroid of R, and V a GLI of index at most n. Then either \(V\subset Z\), [I,R]\(\subset V\) for I a nonzero ideal of R, or \(V\subset C+S\) where \(S\subset RC+C\) and satisfies \(S^{3n-3}=0\). A necessary preliminary result of independent interest shows that when \(V^{(n)}(R)=0\), for V a subset, then the third condition of the theorem holds with the index of nilpotence of S equal to \([(n+1)/2]\).
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 4122166
zbMATH Keywords: inner Lie ideal
zbMATH Keywords: generalized Lie ideal
zbMATH Keywords: prime ring
zbMATH Keywords: center
zbMATH Keywords: extended centroid
zbMATH Keywords: index of nilpotence
author: W. S. III Martindale
author: C. R. Miers
reviewed by: Charles Lanski
MaRDI profile type: Publication
OpenAlex ID: W2052471339
Wikidata QID: Q115366080
cites work: The Lie inner ideal structure of associative rings
cites work: On the Lie structure of an associative ring
cites work: On the iterates of derivations of prime rings
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description: scientific article; zbMATH DE number 4122166