Editing Injective module (section)

==== 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>.