Lemma 10.5.5. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. If $M$ is finite as an $R$-module, then $M$ is finite as an $S$-module.

**Proof.**
In fact, any $R$-generating set of $M$ is also an $S$-generating set of $M$, since the $R$-module structure is induced by the image of $R$ in $S$.
$\square$

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