On Ideals and Commutativity of Prime Rings with Generalized Derivations

Abstract

An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F(xy) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisï¬es certain differential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.