Universally catenary ring
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 universally catenary if every finitely generated algebra over it is a catenary ring.
Since catenary rings are, by definition, Noetherian, so are universally catenary rings.
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| quotient-closed property of commutative unital rings | Yes | universally catenary is quotient-closed | Suppose is a universally catenary ring and is an ideal in . Then, the quotient ring is also a universally catenary ring. |
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Cohen-Macaulay ring | for every ideal, the depth equals the codimension. | click here |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| catenary ring | click here | |||
| Noetherian ring | click here |