Editing Injective module (section)

===Indecomposables===
Every injective submodule of an injective module is a direct summand, so it is important to understand [[indecomposable module|indecomposable]] injective modules, {{harv|Lam|1999|loc=§3F}}.

Every indecomposable injective module has a [[local ring|local]] [[endomorphism ring]]. A module is called a ''[[uniform module]]'' if every two nonzero submodules have nonzero intersection. For an injective module ''M'' the following are equivalent:
* ''M'' is indecomposable
* ''M'' is nonzero and is the injective hull of every nonzero submodule
* ''M'' is uniform
* ''M'' is the injective hull of a uniform module
* ''M'' is the injective hull of a uniform [[cyclic module]]
* ''M'' has a local endomorphism ring

Over a Noetherian ring, every injective module is the direct sum of (uniquely determined) indecomposable injective modules. Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in {{harv|Matlis|1958}}. The indecomposable injective modules are the injective hulls of the modules ''R''/''p'' for ''p'' a prime ideal of the ring ''R''. Moreover, the injective hull ''M'' of ''R''/''p'' has an increasing filtration by modules ''M''<sub>''n''</sub> given by the annihilators of the ideals ''p''<sup>''n''</sup>, and ''M''<sub>''n''+1</sub>/''M''<sub>''n''</sub> is isomorphic as finite-dimensional vector space over the quotient field ''k''(''p'') of ''R''/''p'' to Hom<sub>''R''/''p''</sub>(''p''<sup>''n''</sup>/''p''<sup>''n''+1</sup>, ''k''(''p'')).