Noetherian ring: Difference between revisions
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A subring of a Noetherian ring is not necessarily Noetherian. For this, consider any non-Noetherian [[integral domain]]; this is a subring of a [[field]], which is Noetherian. | A subring of a Noetherian ring is not necessarily Noetherian. For this, consider any non-Noetherian [[integral domain]]; this is a subring of a [[field]], which is Noetherian. | ||
{{localization-closed curing property}} | |||
A localization of a Noetherian ring is Noetherian. Intuitively, when we take localizations, we land up with fewer ideals, so the ascending chain condition becomes easier to satisfy. {{further|[[Noetherianness is localization-closed]]}} | |||
Revision as of 23:36, 7 January 2008
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
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
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Relation with other properties
Stronger properties
Metaproperties
Closure under taking the polynomial ring
This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if
is a commutative unital ring satisfying the property, so is
View other polynomial-closed properties of commutative unital rings
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. Further information: Noetherianness is polynomial-closed
Closure under taking quotient rings
This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this property
View other quotient-closed properties of commutative unital rings
The quotient ring of a Noetherian ring by an ideal, is also Noetherianness. Further information: Noetherianness is quotient-closed
Closure under taking subrings
This property of commutative unital rings is not closed under taking subrings; in other words, a subring of a commutative unital ring with this property need not have this property
A subring of a Noetherian ring is not necessarily Noetherian. For this, consider any non-Noetherian integral domain; this is a subring of a field, which is Noetherian.
Closure under taking localizations
This property of commutative unital rings is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.
View other localization-closed properties of commutative unital rings
A localization of a Noetherian ring is Noetherian. Intuitively, when we take localizations, we land up with fewer ideals, so the ascending chain condition becomes easier to satisfy. Further information: Noetherianness is localization-closed