Editing Artinian module (section)

{{Short description|Module which satisfies the descending chain condition on submodules}}
In [[mathematics]], specifically [[abstract algebra]], an '''Artinian module''' is a [[module (mathematics)|module]] that satisfies the [[descending chain condition]] on its [[poset]] of [[submodule]]s. They are for modules what [[Artinian ring]]s are for [[ring (mathematics)|rings]], and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication).  Both concepts are named for [[Emil Artin]].

In the presence of the axiom of ([[axiom of dependent choice|dependent]]) [[axiom of choice|choice]], the descending chain condition becomes equivalent to the [[minimum condition]], and so that may be used in the definition instead.

Like [[Noetherian module]]s, Artinian modules enjoy the following heredity property:
* If ''M'' is an Artinian ''R''-module, then so is any submodule and any [[quotient module|quotient]] of ''M''.
The [[converse (logic)|converse]] also holds:
* If ''M'' is any ''R''-module and ''N'' any Artinian submodule such that ''M''/''N'' is Artinian, then ''M'' is Artinian.
As a consequence, any [[finitely-generated module]] over an Artinian ring is Artinian.<ref name="Lam-19">Lam (2001), [{{Google books|plainurl=y|id=VtvwJzpWBqUC|page=19|text=Proposition}} Proposition 1.21, p. 19].</ref> Since an Artinian ring is also a [[Noetherian ring]], and finitely-generated modules over a Noetherian ring are Noetherian,<ref name="Lam-19"/> it is true that for an Artinian ring ''R'', any finitely-generated ''R''-module is both Noetherian and Artinian, and is said to be of finite [[length of a module|length]]. It also follows that any finitely generated Artinian module is Noetherian even without the assumption of ''R'' being Artinian. However, if ''R'' is not Artinian and ''M'' is not finitely-generated, [[#Relation to the Noetherian condition|there are counterexamples]].