Lemma 47.3.4. Let $R \to S$ be a ring map. If $E$ is an injective $R$-module, then $\mathop{\mathrm{Hom}}\nolimits _ R(S, E)$ is an injective $S$-module.

**Proof.**
This is true because $\mathop{\mathrm{Hom}}\nolimits _ S(N, \mathop{\mathrm{Hom}}\nolimits _ R(S, E)) = \mathop{\mathrm{Hom}}\nolimits _ R(N, E)$ by Algebra, Lemma 10.13.4.
$\square$

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