Identities related to generalized derivation on ideal in prime rings

Abstract

Let R be a prime ring with center Z(R), I a non-zero ideal of R and $\alpha :R\rightarrow R$ any mapping on R. Suppose that G and F are two generalized derivations associated with derivations g and d respectively on R. In this paper we study the following situations: (i) $G(xy)\pm F(x)F(y)\pm xy \in Z(R)$, (ii) $G(xy)\pm F(y)F(x)\pm xy \in Z(R)$, (iii) $G(xy)\pm F(x)F(y)\pm yx \in Z(R)$, (iv) $G(xy)\pm F(y)F(x) \pm yx \in Z(R)$, (v) $G(xy)\pm F(y)F(x)\pm [x, y] \in Z(R)$, (vi) $G(xy)\pm F(x)F(y) \pm [\alpha (x), y] \in Z(R)$ for all $x, y \in I$. Further an example is given to show that the primeness condition is not superfluous.