Noetherian ring: Difference between revisions
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==Definition | {{commring property}} | ||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
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{{fillin}} | {{fillin}} | ||
== | ==Relation with other properties== | ||
===Stronger properties=== | |||
* [[Polynomial ring over a field]] | |||
* [[Artinian ring]] | |||
* [[Noetherian domain]] | |||
* [[Principal ideal ring]] | |||
* [[Dedekind domain]] | |||
==Metaproperties== | |||
{{poly-closed commring property}} | |||
[[ | The polynomial ring over a Noetherian ring is again Noetherian. This is a general formulation of the [[Hilbert basis theorem]], which asserts in particular that the [[polynomial ring over a field]] is Noetherian. | ||
Revision as of 15:40, 30 June 2007
This article defines a property of commutative rings
Definition
Symbol-free definition
A commutative unital ring is termed Noetherian if it satisfies the following equivalent conditions:
- Ascending chain condition on ideals: Any ascending chain of ideals stabilizes after a finite length
- Every ideal is finitely generated
Definition with symbols
Fill this in later
Relation with other properties
Stronger properties
Metaproperties
Template:Poly-closed commring property
The polynomial ring over a Noetherian ring is again Noetherian. This is a general formulation of the Hilbert basis theorem, which asserts in particular that the polynomial ring over a field is Noetherian.