Lemma 47.5.7. Let $\mathfrak p \subset R$ be a prime of a ring $R$. Let $E$ be the injective hull of $R/\mathfrak p$. Then

1. $E$ is indecomposable,

2. $E$ is the injective hull of $\kappa (\mathfrak p)$,

3. $E$ is the injective hull of $\kappa (\mathfrak p)$ over the ring $R_\mathfrak p$.

Proof. By Lemma 47.2.4 the inclusion $R/\mathfrak p \subset \kappa (\mathfrak p)$ is an essential extension. Then Lemma 47.5.3 shows (2) holds. For $f \in R$, $f \not\in \mathfrak p$ the map $f : \kappa (\mathfrak p) \to \kappa (\mathfrak p)$ is an isomorphism hence the map $f : E \to E$ is an isomorphism, see Lemma 47.5.3. Thus $E$ is an $R_\mathfrak p$-module. It is injective as an $R_\mathfrak p$-module by Lemma 47.3.3. Finally, let $E' \subset E$ be a nonzero injective $R$-submodule. Then $J = (R/\mathfrak p) \cap E'$ is nonzero. After shrinking $E'$ we may assume that $E'$ is the injective hull of $J$ (see Lemma 47.5.3 for example). Observe that $R/\mathfrak p$ is an essential extension of $J$ for example by Lemma 47.2.4. Hence $E' \to E$ is an isomorphism by Lemma 47.5.3 part (3). Hence $E$ is indecomposable. $\square$

Comment #3444 by Sebastian Bozlee on

I don't think that the first sentence establishes (2), since $\kappa(\mathfrak{p}) \to E$ could in principle be non-injective. How about the following?

By Lemma 08XM, $R/\mathfrak{p} \to \kappa(\mathfrak{p})$ is an essential extension. Then Lemma 08Y4 shows (2) holds.

There are also:

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