Noetherian ring: Difference between revisions
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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. | 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|[[Noetherianness is polynomial-closed]]}} | {{further|[[Noetherianness is polynomial-closed]]}} | ||
{{Q-closed curing property}} | |||
The [[quotient ring]] of a Noetherian ring by an ideal, is also Noetherianness. {{further|[[Noetherianness is quotient-closed]]}} | |||
Revision as of 19:18, 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
Fill this in later
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