A note on (α,β)-derivations

We show that every multiplicative (α,β)-derivation of a ring R is additive if there exists an idempotent e′ in R satisfying the conditions (C1)–(C3): (C1) β(e′)Rx=0 implies x=0; (C2) β(e′)xα(e′)R(1-α(e′))=0 implies β(e′)xα(e′)=0; (C3) xR=0 implies x=0. In particular, every multiplicative (α,β)-derivation of a prime ring with a nontrivial idempotent is additive. As applications, we could decompose a multiplicative (α,β)-derivation of the algebra Mn(C) of all the n×n complex matrices into a sum of an (α,β)-inner derivation and an (α,β)-derivation on Mn(C) given by an additive derivation f on C.