Injective hull

Remarks

Every module has an injective hull, and two injective hulls $i \colon M \to E$ and $i' \colon M \to E'$ are related by an isomorphism $f \colon E \to E'$ with $f \circ i = i'$. Equivalently, an injective hull is a maximal essential extension, or a minimal injective module containing $M$. Example: over $R = \Z$, the inclusion $\Z \to \Q$ is an injective hull; $\Q$ is divisible, hence injective, and a nonzero subgroup $N$ of $\Q$ containing $a/b \ne 0$ also contains $b \cdot (a/b) = a \in N \cap \Z$. Non-example: the inclusion $\Z \to \Q \oplus \Q/\Z$, $n \mapsto (n, 0)$, has an injective target but is not essential, since the submodule $0 \oplus \Q/\Z$ meets the image only in $0$. For a prime $p$, the injective hull of $\Z/p\Z$ over $\Z$ is the Prüfer group $\Z[1/p]/\Z$.