##  [Injectivity](/injectivity-0) 

 Definition

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

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

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

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

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

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

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

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

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.