Editing Injective hull (section)

==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>.