injective module

An injective module is the mirror image of a projective module: instead of lifting maps along surjections, it extends maps along injections. Think of it as a module with no obstructions to enlarging the domain — whenever you have a map defined on a submodule and want to extend it to the whole module, an injective target lets you do so. It is a 'complete' or 'divisible enough' module into which things can always be pushed.

Formally, an R-module E is injective if for every injection i : A -> B and every map f : A -> E, there exists g : B -> E with g∘i = f. By Baer's criterion it suffices to check this for inclusions of left ideals into R: E is injective iff every R-homomorphism from a left ideal I into E extends to R -> E. Equivalently, the functor Hom(-, E) is exact, and E is a direct summand of every module containing it.

Over Z (and more generally over a PID), injective is the same as divisible: an abelian group is injective iff for every element x and every nonzero integer n there is y with ny = x. Thus Q and Q/Z are injective Z-modules, while Z itself is not. Every module embeds into an injective one (the injective envelope is the smallest such), which is what makes injective resolutions and the right-derived functors Ext exist.