Generalized reverse derivations on semiprime rings

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.