Lemma 10.7.2. Let $R \to S$ be a finite ring map. Let $M$ be an $S$-module. Then $M$ is finite as an $R$-module if and only if $M$ is finite as an $S$-module.
Proof. One of the implications follows from Lemma 10.5.5. To see the other assume that $M$ is finite as an $S$-module. Pick $x_1, \ldots , x_ n \in S$ which generate $S$ as an $R$-module. Pick $y_1, \ldots , y_ m \in M$ which generate $M$ as an $S$-module. Then $x_ i y_ j$ generate $M$ as an $R$-module. $\square$
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