Difference between revisions of "Noetherian local ring"

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===Stronger properties===
 
===Stronger properties===
  
* [[Regular local ring]]
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{| class="sortable" border="1"
* [[Local Cohen-Macaulay ring]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Local Artinian ring]]
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|-
* [[Local Noetherian domain]]
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| [[Weaker than::regular local ring]] || || || || {{intermediate notions short|Noetherian local ring|regular local ring}}
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|-
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| [[Weaker than::local Cohen-Macaulay ring]] || [[local ring]] that is a [[Cohen-Macaulay ring]]|| || || {{intermediate notions short|Noetherian local ring|local Cohen-Macaulay ring}}
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|-
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| [[Weaker than::local Artinian ring]] || [[local ring]] that is an [[Artinian ring]]|| || || {{intermediate notions short|Noetherian local ring|local Artinian ring}}
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|-
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| [[Weaker than::local Noetherian domain]] || also an [[integral domain]] || || || {{intermediate notions short|Noetherian local ring|local Noetherian domain}}
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|}

Latest revision as of 04:29, 18 July 2013

This article defines a property that can be evaluated for a local ring
View other properties of local rings

Definition

A Noetherian local ring (or local Noetherian ring) is a commutative unital ring that is both a Noetherian ring (i.e. every ideal is finitely generated) and a local ring (i.e. there is a unique maximal ideal).

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
regular local ring click here
local Cohen-Macaulay ring local ring that is a Cohen-Macaulay ring click here
local Artinian ring local ring that is an Artinian ring click here
local Noetherian domain also an integral domain click here