Lemma 15.22.9. Let $R$ be a domain. Any flat $R$-module is torsion free.

**Proof.**
If $x \in R$ is nonzero, then $x : R \to R$ is injective, and hence if $M$ is flat over $R$, then $x : M \to M$ is injective. Thus if $M$ is flat over $R$, then $M$ is torsion free.
$\square$

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