Lemma 10.40.7. Let $R$ be a ring, let $M$ be an $R$-module, and let $m \in M$. Then $\mathfrak p \in V(\text{Ann}(m))$ if and only if $m$ does not map to zero in $M_\mathfrak p$.

**Proof.**
We may replace $M$ by $Rm \subset M$. Then (1) $\text{Ann}(m) = \text{Ann}(M)$ and (2) $x$ does not map to zero in $M_\mathfrak p$ if and only if $\mathfrak p \in \text{Supp}(M)$. The result now follows from Lemma 10.40.5.
$\square$

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