Injective Module over Principal Ideal Domain

Theorem

Let $D$ be a principal ideal domain.

Let $M$ be a $D$-module.

Then $M$ is injective if and only if it is divisible.

Proof

By Principal Ideal Domain is Dedekind Domain $D$ is a Dedekind domain.

By Injective Module over Dedekind Domain $M$ is injective if and only if it is divisible.

$\blacksquare$