On Jordan ideals and left \((\theta,\theta)\)-derivations in prime rings.
type: Item
instance of: scholarly article
title: On Jordan ideals and left \((\theta,\theta)\)-derivations in prime rings.
zbMATH Open document ID: 1069.16041
DOI: 10.1155/S0161171204309075
author: S. M. A. Zaidi
author: Mohammad Ashraf
author: Shakir Ali
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-25T00:00:00Z
full work available at URL: https://eudml.org/doc/51552
review text: Summary: Let \(R\) be a ring and \(S\) a nonempty subset of \(R\). Suppose that \(\theta\) and \(\phi\) are endomorphisms of \(R\). An additive mapping \(\delta\colon R\to R\) is called a left \((\theta,\phi)\)-derivation (resp., Jordan left \((\theta,\phi)\)-derivation) on \(S\) if \(\delta(xy)=\theta(x)\delta(y)+\phi(y)\delta(x)\) (resp., \(\delta(x^2)=\theta(x)\delta(x)+\phi(x)\delta(x)\)) holds for all \(x,y\in S\). Suppose that \(J\) is a Jordan ideal and a subring of a \(2\)-torsion-free prime ring \(R\). In the present paper, it is shown that if \(\theta\) is an automorphism of \(R\) such that \(\delta(x^2)=2\theta(x)\delta(x)\) holds for all \(x\in J\), then either \(J\subseteq Z(R)\) or \(\delta(J)=(0)\). Further, a study of left \((\theta,\theta)\)-derivations of a prime ring \(R\) has been made which act either as a homomorphism or as an antihomomorphism of the ring \(R\).
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 2171564
zbMATH Keywords: endomorphisms
zbMATH Keywords: additive mappings
zbMATH Keywords: Jordan left derivations
zbMATH Keywords: Jordan ideals
zbMATH Keywords: prime rings
zbMATH Keywords: automorphisms
zbMATH Keywords: centers
zbMATH Keywords: homomorphisms
zbMATH Keywords: antihomomorphisms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2171564