Editing Injective module (section)

=== 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" />