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The Stacks project

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


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