##  [Injectivity](/injectivity) 

  ##  [Injectivity](https://algebra.quantumdictionary.io/injectivity-0) 

  

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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.