Editing Noetherian module

In [[abstract algebra]], a '''Noetherian module''' is a [[module (mathematics)|module]] that satisfies the [[ascending chain condition]] on its [[submodule]]s, where the submodules are [[partially ordered]] by [[inclusion (set theory)|inclusion]].<ref>{{harvnb|Roman|2008|loc=p. 133 §5}}</ref>

Historically, [[David Hilbert|Hilbert]] was the first mathematician to work with the properties of [[finitely generated module|finitely generated submodules]]. He [[mathematical proof|proved]] an important theorem known as [[Hilbert's basis theorem]] which says that any [[ideal (ring theory)|ideal]] in the multivariate [[polynomial ring]] of an arbitrary [[field (mathematics)|field]] is [[Ideal (ring theory)#Types of ideals|finitely generated]]. However, the property is named after [[Emmy Noether]] who was the first one to discover the true importance of the property.

==Characterizations and properties==

In the presence of the [[axiom of choice]],<ref>{{Cite web |title=commutative algebra - Is every Noetherian module finitely generated? |url=https://math.stackexchange.com/q/147983 |access-date=2022-05-04 |website=Mathematics Stack Exchange |language=en}}</ref>{{Better source needed|date=May 2022}} two other characterizations are possible: 
*Any [[empty set|nonempty]] set ''S'' of submodules of the module has a [[maximal element]] (with respect to [[set inclusion]]). This is known as the [[maximum condition]].
*All of the submodules of the module are [[finitely generated module|finitely generated]].<ref>{{harvnb|Roman|2008|loc=p. 133 §5 Theorem 5.7}}</ref>

If ''M'' is a module and ''K'' a submodule, then ''M'' is Noetherian [[if and only if]] ''K'' and ''M''/''K'' are Noetherian. This is in contrast to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated.<ref>{{harvnb|Roman|2008|loc=p. 113 §4}}</ref>

==Examples==
*The [[integer]]s, considered as a module over the [[ring (mathematics)|ring]] of integers, is a Noetherian module.
*If ''R''&nbsp;=&nbsp;M<sub>''n''</sub>(''F'') is the full [[matrix ring]] over a field, and ''M''&nbsp;=&nbsp;M<sub>''n'' 1</sub>(''F'') is the set of column vectors over ''F'', then ''M'' can be made into a module using [[matrix multiplication]] by elements of ''R'' on the left of elements of ''M''. This is a Noetherian module.
*Any module that is finite as a set is Noetherian.
*Any finitely generated right module over a right [[Noetherian ring]] is a Noetherian module.

==Use in other structures==
A right [[Noetherian ring]] ''R'' is, by definition, a Noetherian right ''R''-module over itself using multiplication on the right.  Likewise a ring is called left Noetherian ring when ''R'' is Noetherian considered as a left ''R''-module.  When ''R'' is a [[commutative ring]] the left-right adjectives may be dropped as they are unnecessary.  Also, if ''R'' is Noetherian on both sides, it is customary to call it Noetherian and not "left and right Noetherian".
 
The Noetherian condition can also be defined on [[bimodule]] structures as well: a Noetherian bimodule is a bimodule whose [[poset]] of sub-bimodules satisfies the ascending chain condition. Since a sub-bimodule of an ''R''-''S'' bimodule ''M'' is in particular a left ''R''-module, if ''M'' considered as a left ''R''-module were Noetherian, then ''M'' is automatically a Noetherian bimodule. It may happen, however, that a bimodule is Noetherian without its left or right structures being Noetherian.

== See also ==
* [[Artinian module]]
* [[Ascending chain condition|Ascending/descending chain condition]]
* [[Composition series]]
* [[Finitely generated module]]
* [[Krull dimension]]

== References ==
{{Reflist}}
* Eisenbud ''Commutative Algebra with a View Toward Algebraic Geometry'', Springer-Verlag, 1995.

*{{citation | last=Roman | first=Stephen
 | title=Advanced Linear Algebra | edition=Third | series=[[Graduate Texts in Mathematics]] | publisher  = Springer | date=2008| pages= | isbn=978-0-387-72828-5 |author-link=Steven Roman}}

[[Category:Module theory]]
[[Category:Commutative algebra]]

[[de:Emmy Noether#Ehrungen]]