Injective module - Imported Knowlpedia Content

An injective module is a left module $E$ over a ring $R$ such that for every injective module homomorphism $i\colon A\hookrightarrow B$ and every homomorphism $f\colon A\to E$ , there exists a homomorphism $g\colon B\to E$ with $g\circ i=f$ .

Equivalent characterizations

Equivalently, the contravariant functor $\mathrm{Hom}_R(-,E)$ is exact on exact sequences (or, equivalently, $\mathrm{Ext}^1_R(-,E)=0$ ). Injective modules are the categorical dual of projective modules, and they can be recognized by Baer’s criterion in many settings.

Examples

Over a field, every vector space is injective as a module.
As a $\mathbb Z$ -module, $\mathbb Q/\mathbb Z$ is injective (more generally, divisible abelian groups are injective $\mathbb Z$ -modules).
If $\{E_i\}_{i\in I}$ are injective $R$ -modules, then $\prod_{i\in I} E_i$ is injective.