On Jordan ideals and left \((\theta,\theta)\)-derivations in prime rings.
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 \(R\).
scientific article; zbMATH DE number 2171564