Corollary 2

Suppose $A$ is integral. a) Every injective $A$-module is divisible. b) Every torsion-free (II, p. 115) and divisible $A$-module is injective. c) If $A$ is principal, every divisible $A$-module is injective.

If $a \in A$ is nonzero, then $a_A$ is injective; on the other hand, for every $A$-module $E$, the homothety $a_E$ identifies canonically with

therefore $E$ is divisible if and only if $\operatorname{Hom}(a_E, 1_E)$ is surjective for every $a \in A$ nonzero. The assertion a) follows therefore from Definition 2 (X, p. 15).

Let $E$ be a divisible $A$-module; suppose that $A$ is principal (resp. $E$ is torsion-free) and let us prove that $E$ is injective by applying prop. 10. Let $\alpha$ be an ideal of $A$ and $f : \alpha \to E$ an $A$-homomorphism. Let $x \in \alpha$ be such that $\alpha = Ax$ (resp. such that $x \neq 0$ if $\alpha \neq 0$), and let $e \in E$ be such that $xe = f(x)$. Let us prove that for every $a \in \alpha$, we have

this is clear if $a \in Ax$, whence the assertion in the case where A is principal; if E is torsion-free and if $xa \in a$, we have $xf(a) = f(ax) = axe$, hence $f(a) = ae$ since $x$ is nonzero if $a \neq 0$.