On commutativity of prime and semiprime rings with generalized derivations

Abstract

Let $R$ be a prime ring, extended centroid $C$ and $m, n, k \geq1$ are fixed integers. If $R$ admits a generalized derivation $F$ associated with a derivation $d$ such that $(F(x)\circ y)^{m}+(x\circ d(y))^{n}=0$ or $(F(x)\circ_{m} y)^{k} + x\circ_{n} d(y)$=0 for all $x, y \in I$, where $I$ is a nonzero ideal of $R$, then either $R$ is commutative or there exist $b\in U$, Utumi ring of quotient of $R$ such that $F(x)=bx$ for all $x \in R$. Moreover, we also examine the case $R$ is a semiprime ring.