Noetherian local ring
From Commalg
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 |
