A module E over a ring R is injective if every R-linear map from a submodule of any module into E extends to the whole module. Equivalently Hom_R(−,E) is exact or Ext1_R(−,E) = 0; injectives are cogenerators in many module categories and satisfy Baer-type extension criteria.

Principle

Extension-of-maps property: injective modules absorb maps defined on substructures without obstruction, serving as receptacles for extensions and allowing one to test extension problems by mapping into injectives.

Demonstration

As Z-modules, Q/Z is an injective cogenerator: any homomorphism defined on a subgroup of an abelian group extends into Q/Z under the Baer criterion. Over fields, all vector spaces are injective and projective, so extension problems trivialize.

Misapplication

Confusing injectivity with being monomorphic or with projectivity; injective modules need not be free or finitely generated, and treating injectives as duals of projectives without attention to ring context can be misleading.

Consequence

Having enough injectives allows one to compute right-derived functors like Ext by injective resolutions; injective envelopes provide canonical minimal extensions and are central in classification of modules and decomposition theorems.

Reversal

Noninjective modules obstruct extension: there exist maps from submodules that cannot be extended, producing nontrivial extension classes and requiring nontrivial right-derived constructions to measure the failure.

Boundary

Injectivity is categorical and homological and depends heavily on the ambient ring; existence of injective envelopes and enough injectives holds in module categories over rings but may fail in more exotic abelian categories without additional hypotheses.

Semantic Tension

Tension between injective as an algebraic extender and topological uses of 'injective' meaning one-to-one; here injective refers to extension capability, not to injective maps. Also tension with projective: dual notions but asymmetries arise over non-self-dual rings.

Synthesis

Injectivity is the property of modules that every map from a submodule extends to the whole module; it is the homological receptacle notion dual to projectivity and underlies extensions, injective resolutions and envelope constructions.