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

Abstract

We investigate some properties of generalized (α,β) -derivations on semiprime rings. Among some other results, we show that if g is a generalized (α, β) -derivation, with associated (α,β)-derivation δ, on a semiprime ring R such that [g(x), α(x)] = 0 for all x ∈ R, then δ(x)[y, z] = 0 for all x, y, z ∈ R and δ is central. We also show that if α, ν, τ are endomorphisms and β,μ are automorphisms of a semiprime ring R and if R has a generalized (α, β)-derivation g , with associated (α, β)-derivation δ , such that g([μ(x),w(y)]) = [ν(x),w(y)]α,τ , where w : R → R is commutativity preserving, then [y, z]δ(w(p)) = 0 for all y, z, p ∈ R.