Lie ideals and differentiations of semiprime rings
type: Item
instance of: scholarly article
title: Lie ideals and differentiations of semiprime rings
zbMATH Open document ID: 0701.16033
author: O. D. Avraamova
published in: Moscow University Mathematics Bulletin
publication date: 1989-01-01T00:00:00Z
review text: The author announces some results, without proof, most of which extend theorems of \textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] to semi-prime rings. Let \(R\) be a semi-prime ring and let \(U\) be a Lie ideal of \(R\) whose intersection with each nonzero ideal of \(R\) is not central. The first result claims that if \(R\) is 2-torsion free and if \(U\) satisfies a polynomial identity of degree \(n\), then \(R\) satisfies a polynomial identity of degree \(2n\). The other results involve the orthogonal completion \(K(R)\) of \(R\). Two statements are that if \(R\) is 2- torsion free and if \(d_ 1,d_ 2\in\text{Der}(R)\) so that either \(d_ 1d_ 2\in\text{Der}(R)\) or \(d_ 1d_ 2(U)=0\), then \(K(R)=R_ 1\oplus R_ 2\) where each \(R_ i\) is invariant under \(d_ j\) and \(d_ i(R_ i)=0\). Finally, if \(d\in\text{Der}(R)\) and \([d(r),r]\) is central for all \(r\in R\), then \(K(R)=R_ 1\oplus R_ 2\) with \(d(R_ 1)=0\), \(d(R_ 2)\subset R_ 2\), and \(R_ 2\) is commutative.
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 4148359
zbMATH Keywords: semi-prime rings
zbMATH Keywords: Lie ideals
zbMATH Keywords: 2-torsion free rings
zbMATH Keywords: polynomial identities
zbMATH Keywords: orthogonal completions
reviewed by: Charles Lanski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4148359