Derivations involving centralizers

Abstract

Let R be a 2-torsion free semiprime ring,D: R → R be a derivation and α,β are surjective centralizers on R. In this paper,we show that if [[D(x),α(x)],β(x)] = 0 and D(β(x)) = β(D(x)) for all x ∈ R,then D is commuting on R.