Lemma 15.22.6. Let $R$ be a domain. Let $M$ be an $R$-module. Then $M$ is torsion free if and only if $M_\mathfrak m$ is a torsion free $R_\mathfrak m$-module for all maximal ideals $\mathfrak m$ of $R$.

Proof. Omitted. Hint: Use Lemma 15.22.3 and Algebra, Lemma 10.23.1. $\square$

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