Note on nilpotent derivations

Abstract:

Let $R$ be a prime ring with center $Z$. Suppose that $d$ is a derivation on $R$ such that ${d^n}(x) \in Z$ for all $x$, where $n$ is a fixed integer. It is shown that either ${d^n}(x) = 0$ for all $x \in R$ or $R$ is a commutative integral domain. Moreover, the same conclusion holds even if we assume that ${d^n}(x) \in Z$ merely for all $x \in I$, where $I$ is a nonzero ideal of $R$.