Injective module

Remarks

The following are equivalent for an $R$-module $E$:

1. $E$ is injective.
2. Every injective linear map $E \to N$ has a left inverse, that is, it splits.
3. The functor $\Hom_R(-, E)$ takes exact sequences to exact sequences.
4. (Baer's criterion) Every linear map from an ideal $I$ of $R$ to $E$ extends to $R$.

Over a principal ideal domain, a module $E$ is injective exactly when it is divisible: $rE = E$ for every nonzero $r \in R$. The same definition works over any ring, commutative or not.

Example: $\Q$ is an injective $\Z$-module. By Baer's criterion it suffices to extend a map $f \colon n\Z \to \Q$ with $n \ne 0$; with $q = f(n)$, the map $g(x) = xq/n$ on $\Z$ satisfies $g(n) = q$.

Non-example: $\Z$ is not an injective $\Z$-module. Take $i \colon \Z \to \Z$, $x \mapsto 2x$, and $f$ the identity of $\Z$. An extension $g$ would satisfy $1 = f(1) = g(2) = 2g(1)$, which has no solution in $\Z$.