On generalized (α, β)-derivations of semiprime rings

We investigate some properties of generalized (alpha, beta)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (alpha, beta)-derivation, with associated (alpha, beta)-derivation delta, on a semiprime ring R such that [g(x), alpha(x)] = 0 for all x is an element of R, then delta(x)[y, z] = 0 for all x, y, z is an element of R and delta is central. We also show that if alpha, nu, tau are endomorphisms and beta, mu are automorphisms of a semiprime ring R and if R has a generalized (alpha, beta)-derivation g, with associated (alpha, beta)-derivation delta, such that g([mu(x), w(y)]) = [nu(x), w(y)](alpha,tau), where w : R -> R is commutativity preserving, then [y, z]delta(w(p)) = 0 for all y, z, p is an element of R.