Lemma 10.7.3. Suppose that $R \to S$ and $S \to T$ are finite ring maps. Then $R \to T$ is finite.

**Proof.**
If $t_ i$ generate $T$ as an $S$-module and $s_ j$ generate $S$ as an $R$-module, then $t_ i s_ j$ generate $T$ as an $R$-module. (Also follows from Lemma 10.7.2.)
$\square$

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