# Difference between revisions of "Equidimensional ring"

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===Stronger properties=== | ===Stronger properties=== | ||

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+ | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

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+ | | [[Weaker than::local Cohen-Macaulay ring]] || || || || {{intermediate notions short|equidimensional ring|local Cohen-Macaulay ring}} | ||

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+ | | [[Weaker than::local Noetherian domain]] || [[integral domain]] with a unique [[maximal ideal]] || || || {{intermediate notions short|equidimensional ring|local Noetherian domain}} | ||

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## Latest revision as of 04:32, 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 ringsVIEW 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 **equidimensional** if it satisfies both the following conditions:

- All its maximal ideals have the same codimension
- All its minimal prime ideals have the same dimension

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

local Cohen-Macaulay ring | click here | |||

local Noetherian domain | integral domain with a unique maximal ideal | click here |