Exercise 111.7.2. Let $A$ be a ring. Let $M$ be an $A$-module. Let $m \in M$.

Show that $I = \{ a \in A \mid am = 0\} $ is an ideal of $A$.

For a prime $\mathfrak p$ of $A$ show that the image of $m$ in $M_\mathfrak p$ is zero if and only if $I \not\subset \mathfrak p$.

Show that $m$ is zero if and only if the image of $m$ is zero in $M_\mathfrak p$ for all primes $\mathfrak p$ of $A$.

Show that $m$ is zero if and only if the image of $m$ is zero in $M_\mathfrak m$ for all maximal ideals $\mathfrak m$ of $A$.

Show that $M = 0$ if and only if $M_{\mathfrak m}$ is zero for all maximal ideals $\mathfrak m$.

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