Generalized reverse derivations and commutativity of prime rings

summary:Let $R$ be a prime ring with center $Z(R)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d(Z(R))\neq 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F(xy)\pm xy\in Z(R)$, (ii) $F([x,y])\pm [F(x),y]\in Z(R)$, (iii) $F([x,y])\pm [F(x),F(y)]\in Z(R)$, (iv) $F(x\circ y)\pm F(x)\circ F(y)\in Z(R)$, (v) $[F(x),y]\pm [x,F(y)]\in Z(R)$, (vi) $F(x)\circ y\pm x\circ F(y)\in Z(R)$ for all $x,y \in I$, then $R$ is commutative