ON GENERALIZED DERIVATIONS OF PRIME AND SEMIPRIME RINGS

Abstract

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $n$ a fixed positive integer. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F(x \circ y))^{n} = x \circ y$ for all $x,y \in I$, then $R$ is commutative. We also examine the case where $R$ is a semiprime ring.