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{{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 ==
* 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''
* Any [[short exact sequence]] 0 →''Q'' → ''M'' → ''K'' → 0 of left ''R''-modules [[split exact sequence|splits]].
* If ''X'' and ''Y'' are left
::[[
*
Injective right ''R''-modules are defined in complete analogy.
== Examples ==
=== First examples ===
Trivially, the zero module {0} is injective.
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>.
====
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> = 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'') = 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}}
=== Textbooks ===
* {{Citation | last1=Anderson | first1=Frank Wylie | last2=Fuller | first2=Kent R | title=Rings and Categories of Modules | url=https://books.google.com/books?id=PswhrD_wUIkC | access-date=30 July 2016 | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | isbn=978-0-387-97845-1 | year=1992}}
*{{Citation | last1=Enochs | first1=Edgar E. | last2=Jenda | first2=Overtoun M. G. | author2-link=Overtoun Jenda | title=Relative homological algebra | publisher=Walter de Gruyter & Co. | ___location=Berlin | series=de Gruyter Expositions in Mathematics | isbn=978-3-11-016633-0 |mr=1753146 | year=2000 | volume=30 | doi=10.1515/9783110803662}}
*{{Citation | last1=Golan | first1=Jonathan S. | last2=Head | first2=Tom | title=Modules and the structure of rings | publisher=Marcel Dekker | series=Monographs and Textbooks in Pure and Applied Mathematics | isbn=978-0-8247-8555-0 | mr=1201818 | year=1991 | volume=147 | url-access=registration | url=https://archive.org/details/modulesstructure0000gola }}
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 |mr=1653294 | year=1999 | doi=10.1007/978-1-4612-0525-8}}
*{{Citation | last1=Rotman | first1=Joseph J. | title=An introduction to homological algebra | publisher=[[Academic Press]] | ___location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-599250-3 |mr=538169 | year=1979 | volume=85}}
=== Primary sources ===
*{{Citation | last1=Baer | first1=Reinhold | author1-link=Reinhold Baer | title=Abelian groups that are direct summands of every containing abelian group | doi=10.1090/S0002-9904-1940-07306-9 |mr=0002886 | zbl = 0024.14902 |year=1940 | journal=[[Bulletin of the American Mathematical Society]] | volume=46 | pages=800–807 | issue=10| doi-access=free }}
*{{Citation | last1=Chase | first1=Stephen U. | title=Direct products of modules | doi=10.2307/1993382 |mr=0120260 | year=1960 | journal=[[Transactions of the American Mathematical Society]] | volume=97 | pages=457–473 | issue=3 | publisher=American Mathematical Society, Vol. 97, No. 3 | jstor=1993382| doi-access=free }}
*{{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 }}
*{{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]]
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