Editing Injective hull

{{Short description|Notion in abstract algebra}}{{About|the injective hull of a module in algebra|injective hulls of metric spaces, also called tight spans, injective envelopes, or hyperconvex hulls|tight span}}
In [[mathematics]], particularly in [[abstract algebra|algebra]], the '''injective hull''' (or '''injective envelope''') of a [[module (mathematics)|module]] is both the smallest [[injective module]] containing it and the largest [[essential extension]] of it.  Injective hulls were first described in {{harv|Eckmann|Schopf|1953}}.

==Definition==

A [[module (mathematics)|module]] ''E'' is called the '''injective hull''' of a module ''M'', if ''E'' is an [[essential extension]] of ''M'', and ''E'' is [[injective module|injective]]. Here, the base ring is a ring with unity, though possibly non-commutative.

==Examples==

* An injective module is its own injective hull.
* The injective hull of an [[integral domain]] (as a module over itself) is its [[field of fractions]] {{harv|Lam|1999|loc=Example 3.35}}.
* The injective hull of a cyclic ''p''-group (as '''Z'''-module) is a [[Prüfer group]] {{harv|Lam|1999|loc=Example 3.36}}.
* The injective hull of a [[torsion-free abelian group]] <math>A</math> is the [[tensor product of modules|tensor product]] <math>\mathbb Q \otimes_{\mathbb Z} A</math>.
* The injective hull of ''R''/rad(''R'') is Hom<sub>''k''</sub>(''R'',''k''), where ''R'' is a finite-dimensional ''k''-[[algebra (ring theory)|algebra]] with [[Jacobson radical]] rad(''R'') {{harv|Lam|1999|loc=Example 3.41}}.
* A [[simple module]] is necessarily the [[socle (mathematics)|socle]] of its injective hull.
* The injective hull of the residue field of a [[discrete valuation ring]] <math>(R,\mathfrak{m},k)</math> where <math>\mathfrak{m} = x\cdot R</math> is <math>R_x/R</math>.<ref>{{Cite web|url=https://www.math.purdue.edu/~walther/snowbird/inj.pdf|title=Injective Modules|last=Walther|first=Uli|date=|website=|page=11|access-date=}}</ref>
* In particular, the injective hull of <math>\mathbb{C}</math> in <math>(\mathbb{C}[[t]],(t),\mathbb{C})</math> is the module <math>\mathbb{C}((t))/\mathbb{C}[[t]]</math>.

==Properties==
* The injective hull of ''M'' is unique up to isomorphisms which are the identity on ''M'', however the isomorphism is not necessarily unique. This is because the injective hull's map extension property is not a full-fledged [[universal property]]. Because of this uniqueness, the hull can be denoted as ''E''(''M'').
* The injective hull ''E''(''M'') is a maximal [[essential extension]] of ''M'' in the sense that if ''M''⊆''E''(''M'') ⊊''B'' for a module ''B'', then ''M'' is not an essential submodule of ''B''.
* The injective hull ''E''(''M'') is a minimal injective module containing ''M'' in the sense that if ''M''⊆''B'' for an injective module ''B'', then ''E''(''M'') is (isomorphic to) a submodule of ''B''.
* If ''N'' is an essential submodule of ''M'', then ''E''(''N'')=''E''(''M'').
* Every module ''M'' has an injective hull. A construction of the injective hull in terms of homomorphisms Hom(''I'', ''M''), where ''I'' runs through the ideals of ''R'', is given by {{harvtxt|Fleischer|1968}}.
* The dual notion of a [[projective cover]] does ''not'' always exist for a module, however a [[flat cover]] exists for every module. 

===Ring structure===
In some cases, for ''R'' a subring of a self-injective ring ''S'', the injective hull of ''R'' will also have a ring structure.{{sfn|Lam|1999|loc=p. 78–80}} For instance, taking ''S'' to be a full [[matrix ring]] over a field, and taking ''R'' to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right ''R''-module ''R'' is ''S''. For instance, one can take ''R'' to be the ring of all upper triangular matrices. However, it is not always the case that the injective hull of a ring has a ring structure, as an example in {{harv|Osofsky|1964}} shows.

A large class of rings which do have ring structures on their injective hulls are the [[nonsingular ring]]s.{{sfn|Lam|1999|loc=p. 366}} In particular, for an [[integral domain]], the injective hull of the ring (considered as a module over itself) is the [[field of fractions]]. The injective hulls of nonsingular rings provide an analogue of the ring of quotients for non-commutative rings, where the absence of the [[Ore condition]] may impede the formation of the [[classical ring of quotients]]. This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in {{harv|Utumi|1956}}, and the connection to injective hulls was recognized in {{harv|Lambek|1963}}.

==Uniform dimension and injective modules==

An ''R'' module ''M'' has finite [[uniform dimension]] (=''finite rank'') ''n'' if and only if the injective hull of ''M'' is a finite direct sum of ''n'' [[indecomposable module|indecomposable submodules]].

==Generalization==
More generally, let '''C''' be an [[abelian category]]. An [[Object (category theory)|object]] ''E'' is an '''injective hull''' of an object ''M'' if ''M'' → ''E'' is an essential extension and ''E'' is an [[injective object]].

If '''C''' is [[locally small category|locally small]], satisfies [[Abelian category|Grothendieck's axiom AB5]] and has [[enough injectives]], then every object in '''C''' has an injective hull (these three conditions are satisfied by the category of modules over a ring).<ref>Section III.2 of {{harv|Mitchell|1965}}</ref> Every object in a [[Grothendieck category]] has an injective hull.

==See also==

*[[Flat cover]], the dual concept of injective hulls. 
*[[Rational hull]]: This is the analogue of the injective hull when considering a maximal [[rational extension]].

==Notes==
<references/>

==References==
* {{Citation | last1=Eckmann | first1=B. | last2=Schopf | first2=A. | title=Über injektive Moduln | doi=10.1007/BF01899665 |mr=0055978 | year=1953 | journal=Archiv der Mathematik | issn=0003-9268 | volume=4 | pages=75–78 | issue=2}}
* {{Citation|last1=Fleischer|first1=Isidore|title=A new construction of the injective hull|journal=[[Canadian Mathematical Bulletin]]|volume=11|year=1968|pages=19–21|mr=0229680|doi=10.4153/CMB-1968-002-3|doi-access=free}}
*{{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 | volume=189 | doi=10.1007/978-1-4612-0525-8}}
*{{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 | issue=3 | pages=511–528 | doi=10.2140/pjm.1958.8.511 | doi-access=free }}
* Matsumura, H. ''Commutative Ring Theory'', Cambridge studies in advanced mathematics volume 8.
*{{Mitchell TOC}}
*{{Citation | last1=Osofsky | first1=B. L. | authorlink = Barbara L. Osofsky | title=On ring properties of injective hulls |mr=0166227 | year=1964 | journal=[[Canadian Mathematical Bulletin]] | issn=0008-4395 | volume=7 | issue=3 | pages=405–413 | doi=10.4153/CMB-1964-039-3| doi-access=free }}
*{{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}}

==External links==
* [http://planetmath.org/injectivehull injective hull] (PlanetMath article)
* [http://planetmath.org/moduleoffiniterank PlanetMath page on modules of finite rank]
[[Category:Module theory]]