If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$.
The cases $ m \le n$ is trivial. When $m>n$, I don't have much idea how to start. I tried to apply the Cayley-Hamilton theorem but doesn't seem to help much.