A note on [formula omitted]-derivations

Abstract

We show that every multiplicative ( α , β ) -derivation of a ring R is additive if there exists an idempotent e ′ ( e ′ ≠ 0 , 1 ) in R satisfying the conditions (C1)–(C3): (C1) β ( e ′ ) R x = 0 implies x = 0 ; (C2) β ( e ′ ) x α ( e ′ ) R ( 1 - α ( e ′ ) ) = 0 implies β ( e ′ ) x α ( e ′ ) = 0 ; (C3) x R = 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 M n ( C ) of all the n × n complex matrices into a sum of an ( α , β ) -inner derivation and an ( α , β ) -derivation on M n ( C ) given by an additive derivation f on C .