On Lie ideals with generalized $$(\alpha , \alpha )$$-derivations in prime rings

Abstract

Let R be a prime ring and $$\alpha $$ an automorphism on R. An additive mapping F on R is said to be a generalized $$(\alpha , \alpha )$$ -derivation on R if there exists an $$(\alpha , \alpha )$$ -derivation d on R such that $$F(xy)=F(x)\alpha (y)+\alpha (x)d(y)$$ holds for all $$x, y\in R$$ . In this paper our main objective is to study the following identities: (i) $$G(xy) \pm F(x)F(y)\in Z(R);$$ (ii) $$G(xy)\pm F(x)F(y)\pm \alpha (yx) =0;$$ (iii) $$G(xy) \pm F(x)F(y)\pm \alpha (xy)\in Z(R);$$ (iv) $$G(xy)\pm F(x)F(y)\pm \alpha ([x, y]) =0;$$ (v) $$G(xy)\pm F(x)F(y)\pm \alpha (x\circ y)=0;$$ for all x, y in some suitable subset of R, where G and F are two generalized $$(\alpha ,\alpha )$$ -derivations on R