Editing Injective module

{{Short description|Mathematical object in abstract algebra}}
In [[mathematics]], especially in the area of [[abstract algebra]] known as [[module theory]], an '''injective module''' is a [[module (mathematics)|module]] ''Q'' that shares certain desirable properties with the '''Z'''-module '''Q''' of all [[rational number]]s. Specifically, if ''Q'' is a [[submodule]] of some other module, then it is already a [[direct summand]] of that module; also, given a submodule of a module ''Y'', any [[module homomorphism]] from this submodule to ''Q'' can be extended to a homomorphism from all of ''Y'' to ''Q''. This concept is [[Dual (category theory)|dual]] to that of [[projective module]]s. Injective modules were introduced in {{harv|Baer|1940}} and are discussed in some detail in the textbook {{harv|Lam|1999|loc=§3}}.

Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: [[Injective cogenerator]]s are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the [[#Injective resolutions|injective dimension]] and represent modules in the [[derived category]]. [[Injective hull]]s are maximal [[essential extension]]s, and turn out to be minimal injective extensions. Over a [[Noetherian ring]], every injective module is uniquely a direct sum of [[indecomposable module|indecomposable]] modules, and their structure is well understood. An injective module over one ring may be not injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as [[group ring]]s of [[finite group]]s over [[field (mathematics)|field]]s. Injective modules include [[divisible group]]s and are generalized by the notion of [[injective object]]s in [[category theory]].

== 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.

== Examples ==
=== First examples ===

Trivially, the zero module {0} is injective.

Given a [[field (mathematics)|field]] ''k'', every ''k''-[[vector space]] ''Q'' is an injective ''k''-module. Reason: if ''Q'' is a subspace of ''V'', we can find a [[basis of a vector space|basis]] of ''Q'' and extend it to a basis of ''V''. The new extending basis vectors [[linear span|span]] a subspace ''K'' of ''V'' and ''V'' is the internal direct sum of ''Q'' and ''K''. Note that the direct complement ''K'' of ''Q'' is not uniquely determined by ''Q'', and likewise the extending map ''h'' in the above definition is typically not unique.

The rationals '''Q''' (with addition) form an injective abelian group (i.e. an injective '''Z'''-module). The [[factor group]] '''Q'''/'''Z''' and the [[circle group]] are also injective '''Z'''-modules. The factor group '''Z'''/''n'''''Z''' for ''n'' > 1 is injective as a '''Z'''/''n'''''Z'''-module, but ''not'' injective as an abelian group.

=== Commutative examples ===

More generally, for any [[integral domain]] ''R'' with field of fractions ''K'', the ''R''-module ''K'' is an injective ''R''-module, and indeed the smallest injective ''R''-module containing ''R''. For any [[Dedekind domain]], the [[quotient module]] ''K''/''R'' is also injective, and its [[indecomposable module|indecomposable]] summands are the [[localization of a ring|localizations]] <math>R_{\mathfrak{p}}/R</math> for the nonzero [[prime ideal]]s <math>\mathfrak{p}</math>. The [[zero ideal]] is also prime and corresponds to the injective ''K''. In this way there is a 1-1 correspondence between prime ideals and indecomposable injective modules.

A particularly rich theory is available for [[commutative ring|commutative]] [[noetherian ring]]s due to [[Eben Matlis]], {{harv|Lam|1999|loc=§3I}}. Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients ''R''/''P'' where ''P'' varies over the [[prime spectrum]] of the ring. The injective hull of ''R''/''P'' as an ''R''-module is canonically an ''R''<sub>''P''</sub> module, and is the ''R''<sub>''P''</sub>-injective hull of ''R''/''P''. In other words, it suffices to consider [[local ring]]s. The [[endomorphism ring]] of the injective hull of ''R''/''P'' is the [[completion (ring theory)|completion]] <math>\hat R_P</math> of ''R'' at ''P''.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/08Z6|title=Lemma 47.7.5 (08Z6)—The Stacks project|website=stacks.math.columbia.edu|access-date=2020-02-25}}</ref>

Two examples are the injective hull of the '''Z'''-module '''Z'''/''p'''''Z''' (the [[Prüfer group]]), and the injective hull of the ''k''[''x'']-module ''k'' (the ring of inverse polynomials). The latter is easily described as ''k''[''x'',''x''<sup>−1</sup>]/''xk''[''x'']. This module has a basis consisting of "inverse monomials", that is ''x''<sup>−''n''</sup> for ''n'' = 0, 1, 2, …. Multiplication by scalars is as expected, and multiplication by ''x'' behaves normally except that ''x''·1 = 0. The endomorphism ring is simply the ring of [[formal power series]].

=== Artinian examples ===

If ''G'' is a [[finite group]] and ''k'' a field with [[characteristic (algebra)|characteristic]] 0, then one shows in the theory of [[group representation]]s that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the [[group ring|group algebra]] ''kG'' are injective. If the characteristic of ''k'' is not zero, the following example may help.

If ''A'' is a unital [[associative algebra]] over the field ''k'' with finite [[dimension of a vector space|dimension]] over ''k'', then Hom<sub>''k''</sub>(−, ''k'') is a [[duality of categories|duality]] between finitely generated left ''A''-modules and finitely generated right ''A''-modules. Therefore, the finitely generated injective left ''A''-modules are precisely the modules of the form Hom<sub>''k''</sub>(''P'', ''k'') where ''P'' is a finitely generated projective right ''A''-module. For [[Frobenius algebra|symmetric algebras]], the duality is particularly well-behaved and projective modules and injective modules coincide.

For any [[Artinian ring]], just as for [[commutative ring]]s, there is a 1-1 correspondence between prime ideals and indecomposable injective modules. The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its [[injective hull]]. For finite-dimensional algebras over fields, these injective hulls are [[finitely-generated module]]s {{harv|Lam|1999|loc=§3G, §3J}}.

==== Computing injective hulls ====
If <math>R</math> is a Noetherian ring and <math>\mathfrak{p}</math> is a prime ideal, set <math>E = E(R/\mathfrak{p})</math> as the injective hull. The injective hull of <math>R/\mathfrak{p}</math> over the Artinian ring <math>R/\mathfrak{p}^k</math> can be computed as the module <math>(0:_E\mathfrak{p}^k)</math>. It is a module of the same length as <math>R/\mathfrak{p}^k</math>.<ref name=":0">{{Cite book|last=Eisenbud|title=Introduction to Commutative Algebra|pages=624, 625}}</ref> In particular, for the standard graded ring <math>R_\bullet = k[x_1,\ldots,x_n]_\bullet</math> and <math>\mathfrak{p}=(x_1,\ldots, x_n)</math>, <math>E = \oplus_i \text{Hom}(R_i, k)</math> is an injective module, giving the tools for computing the indecomposable injective modules for artinian rings over <math>k</math>.

==== Self-injectivity ====
An Artin local ring <math>(R, \mathfrak{m}, K)</math> is injective over itself if and only if <math>soc(R)</math> is a 1-dimensional vector space over <math>K</math>. This implies every local Gorenstein ring which is also Artin is injective over itself since has a 1-dimensional socle.<ref>{{Cite web|url=https://www.math.purdue.edu/~walther/snowbird/inj.pdf|title=Injective Modules|page=10}}</ref> A simple non-example is the ring <math>R = \mathbb{C}[x,y]/(x^2,xy,y^2)</math> which has maximal ideal <math>(x,y)</math> and residue field <math>\mathbb{C}</math>. Its socle is <math>\mathbb{C}\cdot x \oplus\mathbb{C}\cdot y</math>, which is 2-dimensional. The residue field has the injective hull <math>\text{Hom}_\mathbb{C}(\mathbb{C}\cdot x\oplus\mathbb{C}\cdot y, \mathbb{C})</math>.

=== Modules over Lie algebras ===
For a Lie algebra <math>\mathfrak{g}</math> over a field <math>k</math> of characteristic 0, the category of modules <math>\mathcal{M}(\mathfrak{g})</math> has a relatively straightforward description of its injective modules.<ref>{{Cite web|last=Vogan|first=David|title=Lie Algebra Cohomology|url=http://www-math.mit.edu/~dav/cohom.pdf}}</ref> Using the universal enveloping algebra any injective <math>\mathfrak{g}</math>-module can be constructed from the <math>\mathfrak{g}</math>-module<blockquote><math>\text{Hom}_k(U(\mathfrak{g}), V)</math></blockquote>for some <math>k</math>-vector space <math>V</math>. Note this vector space has a <math>\mathfrak{g}</math>-module structure from the injection<blockquote><math>\mathfrak{g} \hookrightarrow U(\mathfrak{g})</math></blockquote>In fact, every <math>\mathfrak{g}</math>-module has an injection into some <math>\text{Hom}_k(U(\mathfrak{g}), V)</math> and every injective <math>\mathfrak{g}</math>-module is a direct summand of some <math>\text{Hom}_k(U(\mathfrak{g}), V)</math>.

== 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.

== 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.

== References ==
=== Notes ===

{{Reflist}}

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*{{Citation | last1=Papp | first1=Zoltán | title=On algebraically closed modules |mr=0121390 | year=1959 | journal=[[Publicationes Mathematicae Debrecen]] | issn=0033-3883 | volume=6 | pages=311–327}}
*{{Citation | last1=Smith | first1=P. F. | title=Injective modules and prime ideals | doi=10.1080/00927878108822627 |mr=614468 | year=1981 | journal=Communications in Algebra | volume=9 | issue=9 | pages=989–999}}
*{{Citation | last1=Utumi | first1=Yuzo | title=On quotient rings |mr=0078966 | year=1956 | journal=Osaka Journal of Mathematics | issn=0030-6126 | volume=8 | pages=1–18}}
*{{Citation | last1=Vámos | first1=P. | title=Ideals and modules testing injectivity | doi=10.1080/00927878308822975 |mr=733337 | year=1983 | journal=Communications in Algebra | volume=11 | issue=22 | pages=2495–2505}}

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{{DEFAULTSORT:Injective Module}}
[[Category:Homological algebra]]
[[Category:Module theory]]