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Regular local ring: Characters & Measurement

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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Regular local ring topic overview

The analysis highlights Characters and Measurement as prominent areas in the source structure around Regular local ring.

Related topics
61
Source areas
6
Connected nodes
67
Extracted relationships
49
Concept neighborhoods
42
Bridge connections
67

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Overview · 18 topics
Characterizations · 13 topics
Regular ring · 10 topics
Examples · 8 topics
Basic properties · 6 topics
Origin of basic notions · 6 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Characterizations

Examples

Basic properties

Origin of basic notions

Regular ring

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Regular local ring connects Entity context

The extracted context around Regular local ring shows recurring relationship patterns in the source. For example, Regular local ring → A-module, Again, Another, Auslander, Buchsbaum, If, It, Jacobian, Jean-Pierre Serre, Let, Once, Oscar Zariski, Regular, This, Wolfgang Krull, Zariski Another extracted example is Regular local ring → Any, By, Every, For, If, In, Irvin Cohen, Krull, More, Still, These, X1, X2, Xd. Use these groups to spot repeated connection types before inspecting the individual relationships.

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

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

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

Regular local ring relationships Subject–Predicate–Object triples

TTTA extracted 49 structured relationships around Regular local ring. Examples in this analysis include 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 and Regular local ring → is a → unique factorization domain.Every localization. The table shows each extracted connection, where it came from and its confidence.

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

Related concept clusters Concept neighborhoods

The concept neighborhoods around Regular local ring bring nearby vocabulary together. In this analysis, examples include Ring, Regular and Dimension. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Regular local ring
    • Ring
    • Regular
    • Dimension
    • Krull
    • Rings
    • Displaystyle
    • Ideal
    • Field
    • Every
    • Mathfrak
    • Global
    • Prime
  • regular local ring
    • Ring
    • Regular
    • Dimension
    • Krull
    • Rings
    • Displaystyle
    • Field
    • Ideal
    • Prime
    • Every
    • Noetherian
    • Property
  • local ring
    • Ring
    • Regular
    • Dimension
    • Displaystyle
    • Field
    • Krull
    • Rings
    • Ideal
    • Prime
    • Every
    • Noetherian
    • Property
  • generators
    • Maximal
    • Minimal
    • Equal
    • Number
    • Ideal
    • Noetherian
    • Property
    • Mathfrak
    • Krull
    • Zariski
    • Displaystyle
    • Dimension
  • maximal ideal
    • Minimal
    • Displaystyle
    • Equal
    • Ideal
    • Maximal
    • Noetherian
    • Mathfrak
    • Number
    • Prime
    • Ring
    • Dim
    • Property
  • krull dimension
    • Dimension
    • Krull
    • Regular
    • Local
    • Global
    • Power
    • Series
    • Ring
    • Rings
    • Field
    • Finite
    • Formal
  • krull's principal ideal theorem
    • Displaystyle
    • Maximal
    • Minimal
    • Mathfrak
    • Equal
    • Prime
    • Noetherian
    • Ring
    • Dim
    • Number
    • Local
    • Krull
  • regular scheme
    • Ring
    • Dimension
    • Krull
    • Rings
    • Displaystyle
    • Ideal
    • Field
    • Every
    • Global
    • Prime
    • Domain
    • Finite

Connections between topic areas Semantic bridges

For Regular local ring, one of the stronger structural bridges in this analysis connects Regular local ring with Overview. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Regular local ringOverview · splits 49 ⟂ 19
Regular local ringCharacterizations · splits 54 ⟂ 14
Regular local ringRegular ring · splits 57 ⟂ 11
Regular local ringExamples · splits 59 ⟂ 9
Regular local ringBasic properties · splits 61 ⟂ 7
Regular local ringOrigin of basic notions · splits 61 ⟂ 7

Map overview Semantic statistics

Regular local ring

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

Source & methodology

TTTA analyzes the structure around Regular local ring to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Characters & Measurement, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Regular local ring · EN edition · Analysis: TopicsToTalkAbout

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