Editing Injective module (section)

=== 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>.