Zaidi, S. M. A.; Ashraf, Mohammad; Ali, Shakir On Jordan ideals and left \((\theta,\theta)\)-derivations in prime rings. (English) Zbl 1069.16041 Int. J. Math. Math. Sci. 2004, No. 37-40, 1957-1964 (2004). 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 \(R\). Cited in 14 Documents MSC: 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16W20 Automorphisms and endomorphisms Keywords:endomorphisms; additive mappings; Jordan left derivations; Jordan ideals; prime rings; automorphisms; centers; homomorphisms; antihomomorphisms PDFBibTeX XMLCite \textit{S. M. A. Zaidi} et al., Int. J. Math. Math. Sci. 2004, No. 37--40, 1957--1964 (2004; Zbl 1069.16041) Full Text: DOI EuDML