On (α,β-derivations of semiprime rings, II

Let α, β be centralizing automorphisms of a semiprime ring R. Then we show that: (i) If R is 2-torsion free and 3-torsion free and d is an (α, β)-derivation of R such that the mapping x → [d(x),x] is centralizing on R, then d is commuting and d(u)[x,y] = 0 for all x,y,u ∈ R; in particular, d is central, (ii) Let R be 2-torsion free and d,g be (α, β)-derivations of R such that d commutes with both α and β and the mapping x → d²(x) + g(x) is centralizing on R, then d and g are both commuting and d(u)[x,y] = 0 = g(u)[x,y] for all x,y, u ∈ R; in particular d and g are central, (iii) If R admits an (α, β)-derivation d which is strong commutativity-preserving on R, then R is commutative, (iv) An additive mapping d on R is an (α, β)-reverse derivation if and only if it is a central (α, β)-derivation. We also show that if α, β are automorphisms and d an (α, β)-reverse derivation on R which is strong commutativity-preserving, then R is commutative.