Catenary ring: Difference between revisions
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==Definition== | ==Definition== | ||
A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring|Noetherian]] satisfies the following condition: | A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring|Noetherian]] and satisfies the following condition: | ||
If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>. | If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>. | ||
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* [[Noetherian ring]] | * [[Noetherian ring]] | ||
==Metaproperties== | |||
{{Q-closed curing property}} | |||
Revision as of 00:36, 8 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
A commutative unital ring is termed a catenary ring or chain ring or is said to satisfy the Saturated Chain Condition if it is Noetherian and satisfies the following condition:
If is a strictly ascending chain of prime ideals, and is a prime ideal between and , then there is either a prime ideal between and or a prime ideal between and .
Relation with other properties
Stronger properties
- Polynomial ring over a field
- Affine ring over a field: For full proof, refer: Affine implies catenary
- Principal ideal domain
- Universally catenary ring
- Cohen-Macaulay ring
Weaker properties
Metaproperties
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