On multiplicative generalized (a, b)-derivations in prime rings

Abstract

Let R be an associative ring with center Z(R). A mapping F:R → R is called a multiplicative generalized derivation on R if there exists a derivation d on R such that F(xy)=F(x)y+xd(y) for all x,y ∈ R. Let α, β be endomorphisms of R. A mapping F:R → R (not necessarily additive) is said to be a multiplicative generalized (α, β)-derivation if there exists an (α, β)-derivation d on R such that F(xy)=F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In this paper we investigate some identities involving multiplicative generalized (α, β)-derivation in a prime ring R and obtain commutativity of R.