Let $A$ be a ring, $M$ an $A$-module and $p$ a prime ideal of $A$ such that $pM \neq M$. According to my intuition, it is not necessarily true that $p M_p \neq M_p$. Any counterexample?
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Take $M=R/I$ and $\mathfrak p$ a prime ideal of $R$ such that $I\not\subseteq \mathfrak p$. Then $M_{\mathfrak p}=0$, so $\mathfrak pM_{\mathfrak p}=M_{\mathfrak p}$. But if $\mathfrak p+I\neq R$, then $\mathfrak pM\neq M$. (A concrete example: $R=K[X,Y]$, $I=(X)$ and $\mathfrak p=(Y)$.)