Commutativity theorems for *-prime rings with differential identities on Jordan ideals.
type: Item
instance of: scholarly article
title: Commutativity theorems for *-prime rings with differential identities on Jordan ideals.
zbMATH Open document ID: 1272.16025
DOI: 10.5402/2012/729356
published in: ISRN Algebra
publication date: 2013-01-10T00:00:00Z
review text: In a (*-) prime ring \(R\) the authors study special identities for Jordan ideals of \(R\) that involve a derivation of \(R\) and commutators or anti-commutators. Let \(R\) be a 2-torsion free ring with derivation \(d\) and nonzero Jordan ideal \(J\): \(J\) is a subgroup of \((R,+)\) so that for all \(x\in J\) and \(r\in R\), \(x\circ r=xr+rx\in J\). For \(x,y\in R\) set \([x,y]=xy-yx\). When \(R\) has an involution * then \(R\) is *-prime if for any \(a,b\in R\), \(aRb=0=aRb^*\) implies that \(a=0\) or \(b=0\). The main results of the authors assume that \(R\) is *-prime, \(d\) is not zero, \(d*=*d\) as functions on \(R\), and \(J=J^*\). Then \(R\) must be commutative if any of \([d(x),d(y)]=[x,y]\), \([d(x),d(y)]=0\), or \([d(x),d(y)]=d([x,y])\) is an identity on \(J\). Also, for any \(d\), if \(d(x)\circ d(y)=0\) or \(d(x)\circ d(y)=d(x\circ y)\) is an identity for \(J\), then \(d=0\). For any prime \(R\), \(d\), and \(J\) as initially defined above, each of these five identities yields the corresponding statement for \(R\).
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16U80
zbMATH DE Number: 6123892
zbMATH Keywords: differential identities
zbMATH Keywords: derivations
zbMATH Keywords: prime rings
zbMATH Keywords: involutions
zbMATH Keywords: Jordan ideals
zbMATH Keywords: commutativity theorems
author: Abdellah Mamouni
author: Lahcen Oukhtite
author: M. Samman
reviewed by: Charles Lanski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.5402/2012/729356
OpenAlex ID: W2058013096
cites work: On Jordan ideals and left \((\theta,\theta)\)-derivations in prime rings.
cites work: Lie and Jordan Structure in Prime Rings with Derivations
cites work: Commutativity of rings with derivations.
cites work: Commutativity of \(*\)-prime rings with generalized derivations.
cites work: Derivations satisfying certain algebraic identities on Jordan ideals.
cites work: Q3083870
cites work: Posner's second theorem for Jordan ideals in rings with involution.
cites work: Commutativity conditions on derivations and Lie ideals in \(\sigma\)-prime rings.
cites work: Generalized derivations centralizing on Jordanideals of rings with involution
cites work: Q4112071
Wikidata QID: Q58691884
description: scientific article; zbMATH DE number 6123892