Universally catenary ring: Difference between revisions
m (4 revisions) |
No edit summary |
||
| Line 11: | Line 11: | ||
===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::Cohen-Macaulay ring]] || for every [[ideal]], the [[depth]] equals the [[codimension]]. || || || {{intermediate notions short|universally catenary ring|Cohen-Macaulay ring}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::catenary ring]] || || || || {{intermediate notions short|catenary ring|universally catenary ring}} | |||
|- | |||
| [[Stronger than::Noetherian ring]] || || || || {{intermediate notions short|Noetherian ring|universally catenary ring}} | |||
==Metaproperties== | ==Metaproperties== | ||
{{Q-closed curing property}} | {{Q-closed curing property}} | ||
Revision as of 16:08, 18 July 2013
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.
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
MetapropertiesClosure under taking quotient rings
|