On derivations in noncommutative semiprime rings and Banach algebras
Bull. Korean Math. Soc. 2005 Vol. 42, No. 4, 671-678 Printed December 1, 2005
Kyoo-Hong Park Seowon University
Abstract : 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.