Lemma 10.52.7. Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \subset R$ be a multiplicative subset. Then $\text{length}_ R(M) \geq \text{length}_{S^{-1}R}(S^{-1}M)$.

**Proof.**
Any submodule $N' \subset S^{-1}M$ is of the form $S^{-1}N$ for some $R$-submodule $N \subset M$, by Lemma 10.9.15. The lemma follows.
$\square$

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