Catenary ring: Difference between revisions

Revision as of 00:33, 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 satisfies the following condition:

If $P<P_{1}<P_{2}<Q$ is a strictly ascending chain of prime ideals, and $P'$ is a prime ideal between $P$ and $Q$ , then there is either a prime ideal between $P$ and $P'$ or a prime ideal between $P'$ and $Q$ .

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

Noetherian ring

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[commutative unital ring]] is said to be '''catenary''' 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]] 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>.