Differential identities, Lie ideals, and Posner's theorems
type: Item
instance of: scholarly article
title: Differential identities, Lie ideals, and Posner's theorems
zbMATH Open document ID: 0614.16028
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: This paper uses the theory of differential identities to obtain generalizations of two well-known results of \textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)]. A number of such generalizations appear in the literature and the purpose here is to give a uniform treatment which yields essentially all of these, and gives new results as well. Let \(R\) be a prime ring with center \(Z\), extended centroid \(C\), and noncentral Lie ideal \(L\). When \(R\) has an involution, \(^ *\), let \(I=I^ *\) be a nonzero ideal of \(R\) and denote the symmetric and skew-symmetric elements of \(I\) by \(T(I)\) and \(K(I)\), respectively. The first part of the paper shows that if \(d\) is a nonzero derivation of \(R\) so that \([x,x^ d]\in Z\) for all \(x\in L\), then either \(R\) is commutative, or char \(R=2\) and \(R\) satisfies the standard identity \(S_ 4\). Furthermore, if \(R\) has an involution and this same relation holds for all \(x\in T(I)\), or all \(x\in K(I)\), then \(R\) satisfies \(S_ 4\). The generalization of Posner's other theorem is that if \(d\) and \(h\) are nonzero derivations of \(R\) so that \(dh\) is a Lie derivation of \(L\) into \(R\) then char \(R=2\) and either \(R\) satisfies \(S_ 4\), or \(h=dc\) for \(c\in C\). Similar, but less definitive results are obtained when one assumes that \(R\) has an involution and that \(dh\) is either a Lie derivation of \(K(I)\) to \(R\), or a Jordan derivation of \(T(I)\) to \(R\).
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 3993801
zbMATH Keywords: symmetric elements
zbMATH Keywords: differential identities
zbMATH Keywords: prime rings
zbMATH Keywords: center
zbMATH Keywords: extended centroid
zbMATH Keywords: noncentral Lie ideals
zbMATH Keywords: involutions
zbMATH Keywords: skew-symmetric elements
zbMATH Keywords: standard identity
zbMATH Keywords: derivations
zbMATH Keywords: Lie derivations
zbMATH Keywords: Jordan derivations
author: Charles Lanski
MaRDI profile type: Publication
full work available at URL: pjm.1988.134.275
OpenAlex ID: W2033384188
DOI: 10.2140/PJM.1988.134.275
description: scientific article; zbMATH DE number 3993801