Lemma 10.63.11. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Then $\mathop{\mathrm{Spec}}(\varphi )(\text{Ass}_ S(M)) \subset \text{Ass}_ R(M)$.

**Proof.**
If $\mathfrak q \in \text{Ass}_ S(M)$, then there exists an $m$ in $M$ such that the annihilator of $m$ in $S$ is $\mathfrak q$. Then the annihilator of $m$ in $R$ is $\mathfrak q \cap R$.
$\square$

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