Editing Injective module (section)

== Generalizations and specializations ==
=== Injective objects ===

{{Main|injective object}}
One also talks about [[injective object]]s in [[category (mathematics)|categories]] more general than module categories, for instance in [[functor category|functor categories]] or in categories of [[sheaf (mathematics)|sheaves]] of O<sub>''X''</sub>-modules over some [[ringed space]] (''X'',O<sub>''X''</sub>). The following general definition is used: an object ''Q'' of the category ''C'' is injective if for any [[monomorphism]] ''f'' : ''X'' → ''Y'' in ''C'' and any morphism ''g'' : ''X'' → ''Q'' there exists a morphism ''h'' : ''Y'' → ''Q'' with ''hf'' = ''g''.

=== Divisible groups ===

{{Main|divisible group}}
The notion of injective object in the category of abelian groups was studied somewhat independently of injective modules under the term [[divisible group]]. Here a '''Z'''-module ''M'' is injective if and only if ''n''⋅''M'' = ''M'' for every nonzero integer ''n''. Here the relationships between [[flat module]]s, [[pure submodule]]s, and injective modules is more clear, as it simply refers to certain divisibility properties of module elements by integers.

=== Pure injectives ===

{{Main|pure injective module}}
In relative homological algebra, the extension property of homomorphisms may be required only for certain submodules, rather than for all. For instance, a [[pure injective module]] is a module in which a homomorphism from a [[pure submodule]] can be extended to the whole module.