Lemma 110.12.1. Let $R$ be a ring. Let $M$ be an $R$-module which is countable. Then $M$ is a finite $R$-module if and only if $M \otimes _ R R^\mathbf {N} \to M^\mathbf {N}$ is surjective.
Proof. If $M$ is a finite module, then the map is surjective by Algebra, Proposition 10.89.2. Conversely, assume the map is surjective. Let $m_1, m_2, m_3, \ldots $ be an enumeration of the elements of $M$. Let $\sum _{j = 1, \ldots , m} x_ j \otimes a_ j$ be an element of the tensor product mapping to the element $(m_ n) \in M^\mathbf {N}$. Then we see that $x_1, \ldots , x_ m$ generate $M$ over $R$ as in the proof of Algebra, Proposition 10.89.2. $\square$
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