Lemma 10.51.6. 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 10.51.5. 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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