Injective module: Difference between revisions

Revision as of 23:39, 7 June 2023

Revision as of 14:49, 13 August 2023

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Using this criterion, one can show that '''Q''' is an injective [[abelian group]] (i.e. an injective module over '''Z'''). More generally, an abelian group is injective if and only if it is [[divisible module|divisible]]. More generally still: a module over a [[principal ideal ___domain]] is injective if and only if it is divisible (the case of vector spaces is an example of this theorem, as every field is a principal ideal ___domain and every vector space is divisible). Over a general integral ___domain, we still have one implication: every injective module over an integral ___domain is divisible.

Baer's criterion has been refined in many ways {{harv|Golan|Head|1991|p=119}}, including a result of {{harv|Smith|1981}} and {{harv|VamosVámos|1983}} that for a commutative Noetherian ring, it suffices to consider only [[prime ideal]]s ''I''. The dual of Baer's criterion, which would give a test for projectivity, is false in general. For instance, the '''Z'''-module '''Q''' satisfies the dual of Baer's criterion but is not projective.

===Injective cogenerators===

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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''⁺ = Hom_'''Z'''(''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|20012000|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.

===Injective hulls===

{{Main|injective hull}}

The [[injective hull]] of a module is the smallest injective module containing the given one and was described in {{harv|Eckmann|ShopfSchopf|1953}}.

One can use injective hulls to define a minimal injective resolution (see below). If each term of the injective resolution is the injective hull of the cokernel of the previous map, then the injective resolution has minimal length.