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Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A {\displaystyle A} be any Noetherian local ring with unique maximal ideal m {\displaystyle {\mathfrak {m}}} , and suppose a 1 , ⋯ , a n {\displaystyle…

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Overview

Characterizations

Examples

Basic properties

Origin of basic notions

Regular ring

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Regular local ring

Nodes68
Edges67
Triples49
Avg. degree1.97
Density0.029412
Components1

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Regular local ring

Top relations

related to Origin of basic notions · 16
Regular local ring → A-module, Again, Another, Auslander, Buchsbaum, If, It, Jacobian, Jean-Pierre Serre, Let, Once, Oscar Zariski, Regular, This, Wolfgang Krull, Zariski
related to Examples · 14
Regular local ring → Any, By, Every, For, If, In, Irvin Cohen, Krull, More, Still, These, X1, X2, Xd
related to Non-examples · 5
Regular local ring → For, In, Krull, The, Using
related to Characterizations · 4
Regular local ring → If, Its Krull, Noetherian, There
related to Regular ring · 4
Regular local ring → In, Krull, Noetherian, The
is a · 3
Regular local ring → Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension, unique factorization domain.Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular, unique factorization domain.Every localization
related to Basic properties · 3
Regular local ring → Buchsbaum, Every, The Auslander

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Important terminology

regular ring local dimension displaystyle field rings krull ideal noetherian mathfrak every dim variety minimal maximal generators localization nonsingular global

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Regular local ringis aNoetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension0.90text
Regular local ringis aunique factorization domain.Every localization0.90text
Regular local ringis aunique factorization domain.Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular0.90text
Regular local ringrelated to Basic propertiesThe Auslander0.60section
Regular local ringrelated to Basic propertiesBuchsbaum0.60section
Regular local ringrelated to Basic propertiesEvery0.60section
Regular local ringrelated to CharacterizationsThere0.60section
Regular local ringrelated to CharacterizationsIf0.60section
Regular local ringrelated to CharacterizationsNoetherian0.60section
Regular local ringrelated to CharacterizationsIts Krull0.60section
Regular local ringrelated to ExamplesEvery0.60section
Regular local ringrelated to ExamplesThese0.60section

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