Proposition 1

Let $A$ be a noetherian ring. For every prime ideal $\mathfrak{p}$ of $A$, let $e_{\mathfrak{p}} : A/\mathfrak{p} \to I(\mathfrak{p})$ be an injective envelope of $A/\mathfrak{p}$ (A, X, § 1, No. 9).

a) The $A$-modules $I(\mathfrak{p})$ are indecomposable.

b) Let $I$ be an indecomposable injective $A$-module; the set $\mathrm{Ass}(I)$ consists of a single element.

c) The mapping $\mathfrak{p} \mapsto \mathrm{cl}(I(\mathfrak{p}))$ is a bijection of $\mathrm{Spec}(A)$ onto $\mathscr{J}(A)$. The inverse bijection associates to an element $I$ of $\mathscr{J}(A)$ the unique element of $\mathrm{Ass}(I)$.

Let $\mathfrak{p} \in \mathrm{Spec}(A)$; let us prove that the module $I(\mathfrak{p})$ is indecomposable. By A, X, p. 21, Corollary 2, it is enough to prove that if $a$ and $b$ are ideals of $A$ containing $\mathfrak{p}$ and distinct from $\mathfrak{p}$, the ideal $a \cap b$ is distinct from $\mathfrak{p}$; now if $a$ is an element of $a - \mathfrak{p}$ and $b$ an element of $b - \mathfrak{p}$, the product $ab$ belongs to $(a \cap b) - \mathfrak{p}$.

Let $I$ be an indecomposable injective $A$-module, and let $\mathfrak{p}, \mathfrak{q}$ be elements of $\mathrm{Ass}(I)$. Then $I$ contains a submodule $M$ isomorphic to $A/\mathfrak{p}$ and a submodule $N$ isomorphic to $A/\mathfrak{q}$. One has $M \cap N \neq 0$ (A, X, p. 21, Proposition 14); for every nonzero element $x$ of $M \cap N$, one has $\mathfrak{p} = \mathrm{Ann} x = \mathfrak{q}$. Since $\mathrm{Ass}(I)$ is not empty (IV, § 1, No. 1, Corollary 1 of Proposition 2), it consists of a single element $\mathfrak{p}(I)$.

We have thus defined two mappings $\mathfrak{p} \mapsto \mathrm{cl}(I(\mathfrak{p}))$ from $\mathrm{Spec}(A)$ into $\mathscr{J}(A)$ and $I \mapsto \mathfrak{p}(I)$ from $\mathscr{J}(A)$ into $\mathrm{Spec}(A)$; let us prove that these two mappings are reciprocal bijections. Let $\mathfrak{p} \in \mathrm{Spec}(A)$; then $\mathfrak{p}$ belongs to $\mathrm{Ass}(I(\mathfrak{p}))$, and is therefore the unique element of $\mathrm{Ass}(I(\mathfrak{p}))$. Let $I$ be an indecomposable injective $A$-module, and $\mathfrak{p}$ the unique element of $\mathrm{Ass}(I)$; then $I$ is an injective envelope of $A/\mathfrak{p}$ (A, X, p. 21, Prop. 14). This completes the proof of the proposition.