Example

Let $A$ be a principal ideal domain, and let $K$ be its field of fractions. The injective $A$-modules are the divisible $A$-modules (A, X, p. 17, Cor. 2). The $A$-module $K$ is an injective envelope of $A$ (A, X, p. 20, Example 1). Let $p$ be an extremal element of $A$, and $p$ the (maximal) ideal $Ap$; let $e : A/p \longrightarrow K/A_p$ denote the homomorphism which maps the class of an element $a \in A$ to the class of $a/p$. Let us prove that $(K/A_p, e)$ is an injective envelope of $A/p$. The homomorphism $e$ is injective. The $A$-module $K/A_p$ is a quotient of a divisible module, hence is divisible. Let $x$ be a nonzero element of $K/A_p$; it is the class of an element $a/p^n$ of $K$, with $a \in A - \{0\}$ and $n \geqslant 1$ (A, VII, p. 10, th. 2). Then $p^{n-1}x = e(a)$, hence $e^{-1}(Ax) \neq 0$, which proves our assertion.

It then follows from th. 1 that every divisible $A$-module is a direct sum of $A$-modules isomorphic to $K$ or to $K/A_p$ for a maximal (principal) ideal $p$ of $A$.