The Stacks project

Lemma 47.3.7. Let $R$ be a Noetherian ring. A direct sum of injective modules is injective.

Proof. Let $E_ i$ be a family of injective modules parametrized by a set $I$. Set $E = \bigcup E_ i$. To show that $E$ is injective we use Injectives, Lemma 19.2.6. Thus let $\varphi : I \to E$ be a module map from an ideal of $R$ into $E$. As $I$ is a finite $R$-module (because $R$ is Noetherian) we can find finitely many elements $i_1, \ldots , i_ r \in I$ such that $\varphi $ maps into $\bigcup _{j = 1, \ldots , r} E_{i_ j}$. Then we can extend $\varphi $ into $\bigcup _{j = 1, \ldots , r} E_{i_ j}$ using the injectivity of the modules $E_{i_ j}$. $\square$


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