Nilpotency of derivations in prime rings

Abstract:

In 1957, E. C. Posner proved that if $\lambda$ and $\delta$ are derivations of a prime ring R, characteristic $R \ne 2$, then $\lambda \delta = 0$ implies either $\lambda = 0$ or $\delta = 0$. We extend this well-known result by showing that, without any characteristic restriction, $\lambda {\delta ^m} = 0$ implies either $\lambda = 0$ or ${\delta ^{4m - 1}} = 0$. We also prove that ${\lambda ^n}\delta = 0$ implies either ${\delta ^2} = 0$ or ${\lambda ^{12n - 9}} = 0$. In the case where ${\lambda ^n}{\delta ^m} = 0$, we show that if $\lambda$ and $\delta$ commute, then at least one of the derivations must be nilpotent.