Cohen-Macaulay 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

Symbol-free definition

A commutative unital ring is termed Cohen-Macaulay if it satisfies the following equivalent conditions:

• For any maximal ideal, the depth equals the codimension
• For any prime ideal, the depth equals the codimension
• For any ideal, the depth equals the codimension

Metaproperties

Closure under taking the polynomial ring

This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if

$R$

is a commutative unital ring satisfying the property, so is

$R[x]$

View other polynomial-closed properties of commutative unital rings

Local nature

This property of commutative unital rings is local in the following sense: a commutative unital ring satisfies the property iff its localizations at all prime ideals satisfy the property

The property of being a Cohen-Macaulay ring is local in the sense that a commutative unital ring is Cohen-Macaulay if and only if its localizations at all maximal ideals are Cohen-Macaulay, if and only if its localizations at all prime ideals are Cohen-Macaulay.