Left ideals and derivations in semiprime rings.
type: Item
instance of: scholarly article
title: Left ideals and derivations in semiprime rings.
zbMATH Open document ID: 1069.16040
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(R\) be a prime or semiprime ring with center \(Z(R)\), let \(L\) be a nonzero left ideal, and let \(D\) and \(E\) be nonzero derivations on \(R\). For \(S\subseteq R\), denote by \((S)\) the ideal generated by \(S\). The author explores the following conditions, suggested by a classic paper of \textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)]: (i) \(DE\) acts like a derivation on \(L\); (ii) \([D(x),x]\in Z(R)\) for all \(x\in L\); (iii) \(xD(x)\in Z(R)\) for all \(x\in L\); (iv) \(D(x)x-xE(x)\in Z(R)\) for all \(x\in L\). Among other results are the following: (A) If \(R\) is semiprime and 2-torsion-free and satisfies (i), then \(LD(L)\subseteq\text{ann}((E(R)))\) and \(LE(L)\subseteq\text{ann}((D(R)))\); (B) If \(R\) is semiprime and satisfies (iv), then \((D(L))\), \((D(R)L)\), \((LD(R))\) and \((LE(L))\) are all central. This paper is the latest in a long sequence of papers by various authors, all motivated by Posner's theorems; and the author includes a helpful description of earlier results and an extensive bibliography.
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16U80
zbMATH DE Number: 2105244
zbMATH Keywords: derivations
zbMATH Keywords: commutativity theorems
zbMATH Keywords: central ideals
zbMATH Keywords: semiprime rings
zbMATH Keywords: prime rings
author: Charles Lanski
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.09.024
OpenAlex ID: W1972292191
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DOI: 10.1016/J.JALGEBRA.2003.09.024
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description: scientific article; zbMATH DE number 2105244