Lemma 47.3.2. Let $R \to S$ be a flat ring map. If $E$ is an injective $S$-module, then $E$ is injective as an $R$-module.

**Proof.**
This is true because $\mathop{\mathrm{Hom}}\nolimits _ R(M, E) = \mathop{\mathrm{Hom}}\nolimits _ S(M \otimes _ R S, E)$ by Algebra, Lemma 10.14.3 and the fact that tensoring with $S$ is exact.
$\square$

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