The Stacks project

Lemma 10.52.8. Let $R$ be a ring with finitely generated maximal ideal $\mathfrak m$. (For example $R$ Noetherian.) Suppose that $M$ is a finite $R$-module with $\mathfrak m^ n M = 0$ for some $n$. Then $\text{length}_ R(M) < \infty $.

Proof. Consider the filtration $0 = \mathfrak m^ n M \subset \mathfrak m^{n-1} M \subset \ldots \subset \mathfrak m M \subset M$. All of the subquotients are finitely generated $R$-modules to which Lemma 10.52.6 applies. We conclude by additivity, see Lemma 10.52.3. $\square$

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