$M_m = 0$ for all maximal ideals $m \subset R$ implies $M=0$?

Question

Suppose $M$ is an $R$-module, and suppose that the localization of $M$ at $m$, $M_m = 0$ for all maximal ideals $m \subset R$. Show that $M=0.$

Answer 1

Suppose $R$ is commutative unitary.

Let $M$ be a $R$-module such that $M_\mathfrak{m}=0$ for every maximal ideal $\mathfrak m$ and let be $x\in M\setminus \{0\}$. Then $I=\mathrm{Ann}_R(x)$ is a non-zero ideal and it is contained in a maximal ideal $\mathfrak{m}$. Since by hypothesis $x/1=0$, we have that $x$ is annihilated by some element in $A\setminus \mathfrak{m}$, but this is a contradiction since $\mathfrak{m}\supseteq \mathrm{Ann}_R(x)$.