Lemma 47.5.2. Let $R$ be a ring. Any $R$-module has an injective hull.

**Proof.**
Let $M$ be an $R$-module. By More on Algebra, Section 15.55 the category of $R$-modules has enough injectives. Choose an injection $M \to I$ with $I$ an injective $R$-module. Consider the set $\mathcal{S}$ of submodules $M \subset E \subset I$ such that $E$ is an essential extension of $M$. We order $\mathcal{S}$ by inclusion. If $\{ E_\alpha \} $ is a totally ordered subset of $\mathcal{S}$, then $\bigcup E_\alpha $ is an essential extension of $M$ too (Lemma 47.2.3). Thus we can apply Zorn's lemma and find a maximal element $E \in \mathcal{S}$. We claim $M \subset E$ is an injective hull, i.e., $E$ is an injective $R$-module. This follows from Lemma 47.3.5.
$\square$

## 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Your email address will not be published.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 (0)

There are also: