Lemma 10.52.6 (00IY)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.52: Length
Lemma 10.52.6 (cite)

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Lemma 10.52.6. Let $R$ be a ring with maximal ideal $\mathfrak m$ . Suppose that $M$ is an $R$ -module with $\mathfrak m M = 0$ . Then the length of $M$ as an $R$ -module agrees with the dimension of $M$ as a $R/\mathfrak m$ vector space. The length is finite if and only if $M$ is a finite $R$ -module.

Proof. The first part is a special case of Lemma 10.52.5. Thus the length is finite if and only if $M$ has a finite basis as a $R/\mathfrak m$ -vector space if and only if $M$ has a finite set of generators as an $R$ -module. $\square$

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