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 $E_{i_ j}$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #7771 by Ben Moonen on

There are also: