Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a noncommutative semiprime ring. Suppose that there exists a derivation $d: R \to R$ such that for all $x \in R$, either $[[d(x),x],d(x)]=0$ or $\langle\langle d(x),x \rangle, d(x) \rangle=0$. In this case $[d(x),x]$ is nilpotent for all $x \in R$. We also apply the above results to a Banach algebra theory.

Keywords: derivation, semiprime ring, Banach algebra

MSC numbers: 47B47