Lemma 15.22.4. Let $R \to R'$ be a flat homomorphism of domains. If $M$ is a torsion free $R$-module, then $M \otimes _ R R'$ is a torsion free $R'$-module.

**Proof.**
If $M$ is torsion free, then $M \subset M \otimes _ R K$ is injective where $K$ is the fraction field of $R$. Since $R'$ is flat over $R$ we see that $M \otimes _ R R' \to (M \otimes _ R K) \otimes _ R R'$ is injective. Since $M \otimes _ R K$ is isomorphic to a direct sum of copies of $K$, it suffices to see that $K \otimes _ R R'$ is torsion free. This is true because it is a localization of $R'$.
$\square$

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