Noetherian local ring: Difference between revisions

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

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).

Property	Meaning	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

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