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.

Keywords:

Commutator , derivation , Extended Centroid , Generalized Derivation , Prime Ring