Injective module: Difference between revisions

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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 not 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 ==

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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|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.

===Injective resolutions===

Every module ''M'' also has an '''injective [[resolution (algebra)|resolution]]''': an [[exact sequence]] of the form

:0 → ''M'' → ''I''⁰ → ''I''¹ → ''I''² → ...

where the ''I''^''j'' 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''^''n'' is nonzero and ''I''^''i''&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''⁰ → 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''.

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===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.

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{{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_''X''-modules over some [[ringed space]] (''X'',O_''X''). 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 ===

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*{{Citation | last1=Dade | first1=Everett C. | author1-link=Everett C. Dade | title=Localization of injective modules | doi=10.1016/0021-8693(81)90213-1 |mr=617087 | year=1981 | journal=[[Journal of Algebra]] | volume=69 | issue=2 | pages=416–425| doi-access=free }}

*{{Citation | last1=Eckmann | first1=B. | author1-link = Beno Eckmann | last2=Schopf | first2=A. | title=Über injektive Moduln | doi=10.1007/BF01899665 | doi-access=free |mr=0055978 | year=1953 | journal=[[Archiv der Mathematik]] | volume=4 | pages=75–78 | issue=2}}

*{{Citation | last1=Lambek | first1=Joachim | author1-link=Joachim Lambek | title=On Utumi's ring of quotients |mr=0147509 | year=1963 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=15 | pages=363–370 | url=http://www.cms.math.ca/cjm/v15/p363 | doi=10.4153/CJM-1963-041-4 | doi-access=free }}

*{{Citation | last1=Matlis | first1=Eben | author1-link=Eben Matlis | title=Injective modules over Noetherian rings | mr=0099360 | year=1958 | journal=[[Pacific Journal of Mathematics]] | issn=0030-8730 | volume=8 | pages=511–528 | doi=10.2140/pjm.1958.8.511 | doi-access=free }}

*{{Citation | last1=Osofsky | first1=B. L. | author-link = Barbara L. Osofsky | title=On ring properties of injective hulls |mr=0166227 | year=1964 | journal=[[Canadian Mathematical Bulletin]] | issn=0008-4395 | volume=7 | pages=405–413 | doi=10.4153/CMB-1964-039-3| doi-access=free }}