Editing Injective module (section)

== Definition ==

A left module ''Q'' over the [[ring (mathematics)|ring]] ''R'' is injective if it satisfies one (and therefore all) of the following equivalent conditions:
* If ''Q'' is a submodule of some other left ''R''-module ''M'', then there exists another submodule ''K'' of ''M'' such that ''M'' is the [[direct sum of modules|internal direct sum]] of ''Q'' and ''K'', i.e. ''Q'' + ''K'' = ''M'' and ''Q'' ∩ ''K'' = {0}.
* Any [[short exact sequence]] 0 →''Q'' → ''M'' → ''K'' → 0 of left ''R''-modules [[split exact sequence|splits]].
* If ''X'' and ''Y'' are left ''R''-modules, ''f'' : ''X'' → ''Y'' is an [[injective]] module homomorphism and ''g'' : ''X'' → ''Q'' is an arbitrary module homomorphism, then there exists a module homomorphism ''h'' : ''Y'' → ''Q'' such that ''hf'' = ''g'', i.e. such that the following diagram [[commutative diagram|commutes]]:
::[[Image:Injective module.svg|200px|commutative diagram defining injective module Q]]
* The [[contravariant functor|contravariant]] [[Hom functor]] Hom(-,''Q'') from the [[category theory|category]] of left ''R''-modules to the category of [[abelian group]]s is [[exact functor|exact]].

Injective right ''R''-modules are defined in complete analogy.