I was wondering if anyone could give me a hint on how to solve this problem.
If $\mathfrak{p}$ is a prime ideal, and $S = A-\mathfrak{p}$ is the complement of $\mathfrak{p}$ in the ring $A$ then $S^{-1}M$ is denoted by $M_\mathfrak{p}$. Show that the natural map $M \rightarrow \prod M_{\mathfrak{p}}$ of a module $M$ into the direct product of all localizations $M_{\mathfrak{p}}$ where $\mathfrak{p}$ ranges over all maximal ideals, is injective.
I have no idea where to start, so any suggestion with this problem are appreciated, thanks!
WLOG suppose $M\neq 0$ and choose $m$ in $M$ nonzero. Let $I= Ann(m)$. Since $m\neq 0$, this is a proper ideal of $A$. Let $\mathfrak m$ be a maximal ideal of $A$ containing $I$. If $m = 0$ in $M_{\mathfrak m}$, then there exists $a \notin \mathfrak m$ such that $a m = 0$ in $M$; but this contradicts $Ann(m) \subseteq \mathfrak m$. Hence for each nonzero $m \in M$, there exists at least one maximal ideal $\mathfrak m$ such that $m$ is not zero in the localization $M_{\mathfrak m}$; this means precisely that the natural map $M\ \to \prod M_{\mathfrak m}$ is injective.