Generalized reverse derivations on semiprime rings

Abstract

We generalize the notion of reverse derivation by introducing generalized reverse derivations. We define an l-generalized reverse derivation (r-generalized reverse derivation) as an additive mapping F : R → R, satisfying F(xy) = F(y)x + yd(x) (F(xy) = d(y)x + yF(x)) for all x, y ∈ R, where d is a reverse derivation of R. We study the relationship between generalized reverse derivations and generalized derivations on an ideal in a semiprime ring. We prove that if F is an l-generalized reverse (or r-generalized) derivation on a semiprime ring R, then R has a nonzero central ideal.