Lemma 15.22.12. Let $R$ be a domain. Let $M$, $N$ be $R$-modules. If $N$ is torsion free, so is $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$.

**Proof.**
Choose a surjection $\bigoplus _{i \in I} R \to M$. Then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \subset \prod _{i \in I} N$.
$\square$

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