Editing Injective module (section)

== Theory ==

=== Structure theorem for commutative Noetherian rings ===
Over a commutative [[Noetherian ring]] <math>R</math>, every injective module is a direct sum of indecomposable injective modules and every indecomposable injective module is the injective hull of the residue field at a prime <math>\mathfrak{p}</math>. That is, for an injective <math>I \in \text{Mod}(R)</math> , there is an isomorphism<blockquote><math>I \cong \bigoplus_{i} E(R/\mathfrak{p}_i)</math></blockquote>where <math>E(R/\mathfrak{p}_i)</math> are the injective hulls of the modules <math>R/\mathfrak{p}_i</math>.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/08YA|title=Structure of injective modules over Noetherian rings}}</ref> In addition, if <math>I</math> is the injective hull of some module <math>M</math> then the <math>\mathfrak{p}_i</math> are the associated primes of <math>M</math>.<ref name=":0" />

=== Submodules, quotients, products, and sums, Bass-Papp Theorem===

Any [[product (category theory)|product]] of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective {{harv|Lam|1999|p=61}}. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite [[direct sum of modules|direct sums]] of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is [[Artinian ring|Artinian]] [[semisimple ring|semisimple]] {{harv|Golan|Head|1991|p=152}}; every factor module of every injective module is injective if and only if the ring is [[hereditary ring|hereditary]], {{harv|Lam|1999|loc=Th. 3.22}}.

Bass-Papp Theorem states that every infinite direct sum of right (left) injective modules is injective if and only if the ring is right (left) [[Noetherian ring|Noetherian]], {{harv|Lam|1999|p=80-81|loc=Th 3.46}}.<ref>This is the [[Hyman Bass|Bass]]-Papp theorem, see {{harv|Papp|1959}} and {{harv|Chase|1960}}</ref>

===Baer's criterion===

In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left ''R''-module ''Q'' is injective if and only if any homomorphism ''g'' : ''I'' → ''Q'' defined on a [[ideal (ring theory)|left ideal]] ''I'' of ''R'' can be extended to all of ''R''.

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|Vámos|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===
{{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.

===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|Schopf|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.

===Injective resolutions===
Every module ''M'' also has an injective [[resolution (algebra)|resolution]]: an [[exact sequence]] of the form
:0 → ''M'' → ''I''<sup>0</sup> → ''I''<sup>1</sup> → ''I''<sup>2</sup> → ...
where the ''I''<sup> ''j''</sup> are injective modules. Injective resolutions can be used to define [[derived functor]]s such as the [[Ext functor]].

The ''length'' of a finite injective resolution is the first index ''n'' such that ''I''<sup>''n''</sup> is nonzero and ''I''<sup>''i''</sup>&nbsp;=&nbsp;0 for ''i'' greater than ''n''. If a module ''M'' admits a finite injective resolution, the minimal length among all finite injective resolutions of ''M'' is called its injective dimension and denoted id(''M''). If ''M'' does not admit a finite injective resolution, then by convention the injective dimension is said to be infinite. {{harv|Lam|1999|loc=§5C}} As an example, consider a module ''M'' such that id(''M'')&nbsp;=&nbsp;0. In this situation, the exactness of the sequence 0 → ''M'' → ''I''<sup>0</sup> → 0 indicates that the arrow in the center is an isomorphism, and hence ''M'' itself is injective.<ref>A module isomorphic to an injective module is of course injective.</ref>

Equivalently, the injective dimension of ''M'' is the minimal integer (if there is such, otherwise ∞) ''n'' such that Ext{{su|p=''N''|b=''A''}}(–,''M'') = 0 for all ''N'' > ''n''.

===Indecomposables===
Every injective submodule of an injective module is a direct summand, so it is important to understand [[indecomposable module|indecomposable]] injective modules, {{harv|Lam|1999|loc=§3F}}.

Every indecomposable injective module has a [[local ring|local]] [[endomorphism ring]]. A module is called a ''[[uniform module]]'' if every two nonzero submodules have nonzero intersection. For an injective module ''M'' the following are equivalent:
* ''M'' is indecomposable
* ''M'' is nonzero and is the injective hull of every nonzero submodule
* ''M'' is uniform
* ''M'' is the injective hull of a uniform module
* ''M'' is the injective hull of a uniform [[cyclic module]]
* ''M'' has a local endomorphism ring

Over a Noetherian ring, every injective module is the direct sum of (uniquely determined) indecomposable injective modules. Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in {{harv|Matlis|1958}}. The indecomposable injective modules are the injective hulls of the modules ''R''/''p'' for ''p'' a prime ideal of the ring ''R''. Moreover, the injective hull ''M'' of ''R''/''p'' has an increasing filtration by modules ''M''<sub>''n''</sub> given by the annihilators of the ideals ''p''<sup>''n''</sup>, and ''M''<sub>''n''+1</sub>/''M''<sub>''n''</sub> is isomorphic as finite-dimensional vector space over the quotient field ''k''(''p'') of ''R''/''p'' to Hom<sub>''R''/''p''</sub>(''p''<sup>''n''</sup>/''p''<sup>''n''+1</sup>, ''k''(''p'')).

===Change of rings===
It is important to be able to consider modules over [[subring]]s or [[quotient ring]]s, especially for instance [[polynomial ring]]s. In general, this is difficult, but a number of results are known, {{harv|Lam|1999|p=62}}.

Let ''S'' and ''R'' be rings, and ''P'' be a left-''R'', right-''S'' [[bimodule]] that is [[flat module|flat]] as a left-''R'' module. For any injective right ''S''-module ''M'', the set of [[module homomorphism]]s Hom<sub>''S''</sub>( ''P'', ''M'' ) is an injective right ''R''-module. The same statement holds of course after interchanging left- and right- attributes.

For instance, if ''R'' is a subring of ''S'' such that ''S'' is a flat ''R''-module, then every injective ''S''-module is an injective ''R''-module. In particular, if ''R'' is an integral domain and ''S'' its [[field of fractions]], then every vector space over ''S'' is an injective ''R''-module. Similarly, every injective ''R''[''x'']-module is an injective ''R''-module.

In the opposite direction, a ring homomorphism <math>f: S\to R</math> makes ''R'' into a left-''R'', right-''S'' bimodule, by left and right multiplication. Being [[free module|free]] over itself ''R'' is also [[flat module#Free and projective modules|flat]] as a left ''R''-module. Specializing the above statement for ''P = R'', it says that when ''M'' is an injective right ''S''-module the [[coinduced module]] <math> f_* M = \mathrm{Hom}_S(R, M)</math> is an injective right ''R''-module. Thus, coinduction over ''f'' produces injective ''R''-modules from injective ''S''-modules.

For quotient rings ''R''/''I'', the change of rings is also very clear. An ''R''-module is an ''R''/''I''-module precisely when it is annihilated by ''I''. The submodule ann<sub>''I''</sub>(''M'') = { ''m'' in ''M'' : ''im'' = 0 for all ''i'' in ''I'' } is a left submodule of the left ''R''-module ''M'', and is the largest submodule of ''M'' that is an ''R''/''I''-module. If ''M'' is an injective left ''R''-module, then ann<sub>''I''</sub>(''M'') is an injective left ''R''/''I''-module. Applying this to ''R''='''Z''', ''I''=''n'''''Z''' and ''M''='''Q'''/'''Z''', one gets the familiar fact that '''Z'''/''n'''''Z''' is injective as a module over itself. While it is easy to convert injective ''R''-modules into injective ''R''/''I''-modules, this process does not convert injective ''R''-resolutions into injective ''R''/''I''-resolutions, and the homology of the resulting complex is one of the early and fundamental areas of study of relative homological algebra.

The textbook {{harv|Rotman|1979|p=103}} has an erroneous proof that [[localization of a ring|localization]] preserves injectives, but a counterexample was given in {{harv|Dade|1981}}.

===Self-injective rings===
Every ring with unity is a [[free module]] and hence is a [[projective module|projective]] as a module over itself, but it is rarer for a ring to be injective as a module over itself, {{harv|Lam|1999|loc=§3B}}. If a ring is injective over itself as a right module, then it is called a right self-injective ring. Every [[Frobenius algebra]] is self-injective, but no [[integral domain]] that is not a [[field (mathematics)|field]] is self-injective. Every proper [[quotient ring|quotient]] of a [[Dedekind domain]] is self-injective.

A right [[Noetherian ring|Noetherian]], right self-injective ring is called a [[quasi-Frobenius ring]], and is two-sided [[Artinian ring|Artinian]] and two-sided injective, {{harv|Lam|1999|loc=Th. 15.1}}.  An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.