Catenary ring: Difference between revisions
| Line 11: | Line 11: | ||
===Stronger properties=== | ===Stronger properties=== | ||
* [[Affine ring]]: {{proofat|[[Affine implies catenary]]}} | * [[Polynomial ring over a field]] | ||
* [[Affine ring over a field]]: {{proofat|[[Affine implies catenary]]}} | |||
* [[Principal ideal domain]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Noetherian ring]] | * [[Noetherian ring]] | ||
Revision as of 00:00, 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 said to be catenary if it is Noetherian 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