Let $R$ be a ring, $P\subset R$ a prime ideal, and $M$ a finitely generated $R$-module. Suppose that $M/PM$ is the zero module over $R/P$. I want to show that $M_P$, i.e., the localization of the module $M$ at the prime $P$, is the zero module over the ring $A_P$, i.e., the localization of $A$ at $P$.
If $(R,P)$ were a local ring, then I would be able to apply Nakayama's lemma to get $M=0$ (since $PM=M$ for $P$ maximal) and hence $M_P=0$. However $R$ is not local. How do I proceed?