Editing Injective module (section)

== 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>.