Lemma 10.52.7 (00IZ)—The Stacks project

Processing math: 100%

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. $\square$

Comments (0)

There are also:

2 comment(s) on Section 10.52: Length

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment