Let $R$ be a prime ring with extended centroid $C$, $H$ a generalized derivationof $R$ and $n\geq 1$ a fixed integer. In this paper we study the situations: (1) If $(H(xy))^n =(H(x))^n(H(y))^n$ for all $x,y\in R$; (2) obtain some related result in case $R$ is a noncommutativeBanach algebra and $H$ is continuous or spectrally bounded. Manuscript profile

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,y\in R$ where $0\neq b\in R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. We also obtain some related result in case $R$ is a non-commutative Banach algebra and d continuous or spectrally bounded. Manuscript profile