On ideals and commutativity of prime rings with generalized derivations

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 is an element of R. It is called a generalized (alpha, beta) derivation on R if there exists an (alpha,beta) derivation d: R -> R such that the equation F(xy) = F(x)alpha(y),beta(x)d(y) holds for all x, y is an element of R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on the left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.