Injective module

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 $g\circ i=f$.

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.