Editing Noetherian ring (section)

== Implication on injective modules ==
Given a ring, there is a close connection between the behaviors of [[injective module]]s over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring ''R'', the following are equivalent:
*''R'' is a left Noetherian ring.
*(Bass) Each direct sum of injective left ''R''-modules is injective.<ref name="Bass injective" />
*Each injective left ''R''-module is a direct sum of [[indecomposable module|indecomposable]] injective modules.<ref>{{harvnb|Anderson|Fuller|1992|loc=Theorem 25.6. (b)}}</ref>
*(Faith–Walker) There exists a [[cardinal number]] <math>\mathfrak{c}</math> such that each injective left module over ''R'' is a direct sum of <math>\mathfrak{c}</math>-generated modules (a module is <math>\mathfrak{c}</math>-generated if it has a [[generating set of a module|generating set]] of [[cardinality]] at most <math>\mathfrak{c}</math>).<ref>{{harvnb|Anderson|Fuller|1992|loc=Theorem 25.8.}}</ref>
*There exists a left ''R''-module ''H'' such that every left ''R''-module [[embedding|embeds]] into a direct sum of copies of ''H''.<ref>{{harvnb|Anderson|Fuller|1992|loc=Corollary 26.3.}}</ref>
<!--Expand this later: Over a commutative ring, decomposing an injective module is essentially the same as doing a primary decomposition and that explains "Noetherian" assumption. -->

The [[endomorphism ring]] of an indecomposable injective module is [[local ring|local]]<ref>{{harvnb|Anderson|Fuller|1992|loc=Lemma 25.4.}}</ref> and thus [[Azumaya's theorem]] says that, over a left Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the [[Krull–Schmidt theorem]]).