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 $F\colon R\to R$, satisfying $F(xy)=F(y)x+yd(x)$ ($F(xy)=d(y)x+yF(x)$) for all $x,y\in R$, 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.