On Jordan ideals and left \((\theta,\theta)\)-derivations in prime rings. (English) Zbl 1069.16041

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 \(R\).


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
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