Noetherian local ring

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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