Lie ideals with regular and nilpotent elements and a result on derivations
type: Item
instance of: scholarly article
title: Lie ideals with regular and nilpotent elements and a result on derivations
zbMATH Open document ID: 0538.16032
DOI: 10.1007/BF02844414
author: Jeffrey Bergen
publication date: 1984-01-01T00:00:00Z
review text: The results in this paper determine the structure of semi-prime rings if either the image of a derivation, or a Lie ideal, has each element satisfying certain conditions. For example, if R is a ring with no nonzero nil left ideal and if d is a derivation of R with every value, d(r), either invertible or nilpotent, then either \(d=0\), R is a division ring, or \(R=M_ 2(D)\) for D a division ring. The same conclusion holds if one assumes only that R is a semi-prime ring but insists that the indices of nilpotence of the d(r) are bounded. To prove these theorems, the author makes use of the known results when all values d(r) are nilpotent [\textit{B. Felzenszwalb} and \textit{C. Lanski}, J. Algebra 83, 520- 530 (1983; Zbl 0519.16022); and \textit{A. Giambruno} and \textit{I. N. Herstein}, Rend. Circ. Mat. Palermo, II. Ser. 30, 199-206 (1981; Zbl 0482.16030)] in order to reduce to the case of a primitive ring, at which point a clever computation using the Jacobson density theorem finishes the proof. Similar theorems are obtained for Lie ideals of R. Specifically, if U is a Lie ideal of R having each of its elements nilpotent or regular, then if R has no nonzero nil left ideal, either U is central, R is a domain, or R is an order in \(M_ 2(F)\) for F a field. The same conclusion holds if one bounds the indices of nilpotence of the elements of U and assumes only that R is a semi-prime ring. The proofs of these results make use of the theorems above for derivations.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 3855335
zbMATH Keywords: semi-prime rings
zbMATH Keywords: derivation
zbMATH Keywords: nil left ideal
zbMATH Keywords: invertible
zbMATH Keywords: nilpotent
zbMATH Keywords: division ring
zbMATH Keywords: indices of nilpotence
zbMATH Keywords: values
zbMATH Keywords: primitive ring
zbMATH Keywords: Jacobson density theorem
zbMATH Keywords: Lie ideals
zbMATH Keywords: regular
zbMATH Keywords: order
reviewed by: Charles Lanski
MaRDI profile type: Publication
cites work: Derivations with Invertible Values
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cites work: JORDAN ALGEBRAS OF CAPACITY TWO
cites work: General polynomial identities. II
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full work available at URL: https://doi.org/10.1007/bf02844414
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3855335