A note on (α, β)-derivations

We show that every multiplicative (alpha, beta)-derivation of a ring R is additive if there exists an idempotent e' (e' not equal 0, 1) in R. satisfying the conditions (C1)-(C3): (C1) beta(e')Rx = 0 implies x = 0; (C2) beta(e')x alpha(e')R(1 - alpha(e')) = 0 implies beta(e')x alpha(e') = 0; (C3)xR = 0 implies x = 0. In particular, every multiplicative (alpha, beta)-derivation of a prime ring with a nontrivial idempotent is additive. As applications, we could decompose a multiplicative (alpha, beta)-derivation of the algebra M-n(C) of all the n x n complex matrices into a sum of an (alpha, beta)-inner derivation and an (alpha, beta)-derivation on M-n (C) given by an additive derivation f on C. (C) 2009 Elsevier Inc. All rights reserved.