Remark 1

Let $I$ be an indecomposable injective $A$-module, and let $\mathfrak{p}$ be the unique element of $\mathrm{Ass}(I)$; by Prop. 1, $I$ contains a submodule isomorphic to $A/\mathfrak{p}$ of which it is the injective envelope. In general such a submodule is not unique, as is seen by taking $A = \mathbf{Z}$, $\mathfrak{p} = 0$, $I = \mathbf{Q}$.

For each prime ideal $\mathfrak{p}$ of $A$, let us choose as above an injective envelope $(I(\mathfrak{p}), e_{\mathfrak{p}})$ of $A/\mathfrak{p}$. By A, X, p. 22, Th. 3, one has: