Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a prime ring of characteristic different from $2$, $C$ the extended centroid of $R$, and $\delta$ a generalized derivations of $R$. If $[[\delta(x),x],\delta(x)]=0$ for all $x\in R$ then either $R$ is commutative or $\delta(x)=ax$ for all $x\in R$ and some $a \in C$. We also obtain some related result in case $R$ is a Banach algebra and $\delta$ is either continuous or spectrally bounded.

Keywords: prime ring, derivations, differential identities, Banach algebras

MSC numbers: Primary 16N60; Secondary 16W25