Editing Injective module (section)

===Injective cogenerators===
{{Main|injective cogenerator}}
Maybe the most important injective module is the abelian group '''Q'''/'''Z'''. It is an [[injective cogenerator]] in the [[category of abelian groups]], which means that it is injective and any other module is contained in a suitably large product of copies of '''Q'''/'''Z'''. So in particular, every abelian group is a subgroup of an injective one. It is quite significant that this is also true over any ring: every module is a submodule of an injective one, or "the category of left ''R''-modules has enough injectives." To prove this, one uses the peculiar properties of the abelian group '''Q'''/'''Z''' to construct an injective cogenerator in the category of left ''R''-modules.

For a left ''R''-module ''M'', the so-called "character module" ''M''<sup>+</sup> = Hom<sub>'''Z'''</sub>(''M'','''Q'''/'''Z''') is a right ''R''-module that exhibits an interesting duality, not between injective modules and [[projective module]]s, but between injective modules and [[flat module]]s {{harv|Enochs|Jenda|2000|pp=78–80}}. For any ring ''R'', a left ''R''-module is flat if and only if its character module is injective. If ''R'' is left noetherian, then a left ''R''-module is injective if and only if its character module is flat.