DERIVATIONS AND CENTRALIZING MAPPINGS IN PRIME RINGS

Abstract

Let $R$ be a prime ring with extended centroid $C$ and $\rho$ a nonzero right ideal of $R$. In this paper we investigate the derivations $\delta$, $d$ on $R$ such that $[\delta (x), d(x)] \in C$ for all $x \in \rho$. As an application, we prove that any centralizing additive mapping $f$ on $\rho$ must be of the form $f(x)=\lambda x+\mu (x)$ for all $x\in\rho$, where $\lambda \in C$ and $\mu : \rho\to C$, except when $[\rho, \rho ]\rho =0$.