Lemma 15.55.8. Let $R$ be a ring. For every $R$-module $M$ the $R$-module $J(M)$ is injective.

**Proof.**
Note that $J(M) \cong \prod _{\varphi \in M^\vee } R^\vee $ as an $R$-module. As the product of injective modules is injective, it suffices to show that $R^\vee $ is injective. For this we use that

and the fact that $(-)^\vee $ is an exact functor by Lemma 15.55.6. $\square$

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## Comments (2)

Comment #113 by Kiran Kedlaya on

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