Results on generalized derivations in prime rings

A prime ring ${S}$ with the centre ${Z}$ and generalised derivations that meet certain algebraic identities is considered. Let's assume that $\Psi$ and $\Phi$ are two generalised derivations associated with $\psi$ and $\phi$ on ${S},$ respectively. In this article, we examine the following identities: (i) $\Psi(a)b-a\Phi(b)\in {Z},$ (ii) $\Psi(a)b-b\Phi(a)\in {Z},$ (iii) $\Psi(a)a-b\Phi(b)\in {Z},$ (iv) $\Psi(a)a-a\Phi(b)\in {Z},$ (v) $\Psi(a)a-b\Phi(a)\in {Z},$ for every $a, b\in {J},$ where ${J}$ is a non-zero two sided ideal of ${S}.$ We also provide an example to show that the condition of primeness imposed in the hypotheses of our results is essential.