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Krull dimension: Examples, Overview & Explanation

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of…

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Krull dimension topic overview

The analysis highlights Examples, Overview and Explanation as prominent areas in the source structure around Krull dimension.

Related topics
54
Source areas
5
Connected nodes
76
Extracted relationships
40
Concept neighborhoods
32
Bridge connections
76

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 · 19 topics
Examples · 16 topics
Explanation · 9 topics
Of a module · 5 topics
Schemes · 5 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

Explanation

Schemes

Examples

Of a module

Bibliography

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 Krull dimension connects Entity context

The extracted context around Krull dimension shows recurring relationship patterns in the source. For example, Krull dimension → An, Artinian, Cohen, Dedekind, For, Given, If, In, Krull, Let, Macaulay, More, Neumann, Noether, Noetherian, The, The Krull, Then Another extracted example is Krull dimension → Given, In, Krull, That, The, The Krull, We. Use these groups to spot repeated connection types before inspecting the individual relationships.

Krull dimension

Top relations

related to Examples · 18
Krull dimension → An, Artinian, Cohen, Dedekind, For, Given, If, In, Krull, Let, Macaulay, More, Neumann, Noether, Noetherian, The, The Krull, Then
related to Explanation · 7
Krull dimension → Given, In, Krull, That, The, The Krull, We
related to Schemes · 6
Krull dimension → Galois, It, Krull, Spec, This, Zariski
related to Of a module · 5
Krull dimension → Ann, End, If, Krull, That
related to For non-commutative rings · 4
Krull dimension → For, Noetherian, The, The Krull

Important terminology

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

Important terminology

dimension ring krull noetherian ideal prime displaystyle ideals rings field height commutative algebra chains supremum isbn mathfrak zero lengths algebraic

Krull dimension relationships Subject–Predicate–Object triples

TTTA extracted 40 structured relationships around Krull dimension. Examples in this analysis include Krull dimension → related to Examples → The and Krull dimension → related to Examples → In. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Krull dimensionrelated to ExamplesThe0.60section
Krull dimensionrelated to ExamplesIn0.60section
Krull dimensionrelated to ExamplesNoetherian0.60section
Krull dimensionrelated to ExamplesIf0.60section
Krull dimensionrelated to ExamplesFor0.60section
Krull dimensionrelated to ExamplesGiven0.60section
Krull dimensionrelated to ExamplesMore0.60section
Krull dimensionrelated to ExamplesAn0.60section
Krull dimensionrelated to ExamplesKrull0.60section
Krull dimensionrelated to ExamplesDedekind0.60section
Krull dimensionrelated to ExamplesThe Krull0.60section
Krull dimensionrelated to ExamplesArtinian0.60section

Related concept clusters Concept neighborhoods

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

  • Krull dimension
    • Krull
    • Ring
    • Field
    • Noetherian
    • Defined
    • Supremum
    • Zero
    • Possibly
    • Ideals
    • Lengths
    • Displaystyle
    • Prime
  • krull dimension
    • Krull
    • Ring
    • Field
    • Noetherian
    • Defined
    • Supremum
    • Zero
    • Possibly
    • Rings
    • Ideals
    • Lengths
    • Displaystyle
  • commutative algebra
    • Rings
    • Commutative
    • Generated
    • Noetherian
    • Mathematics
    • Isbn
    • Krull
    • Zero
    • Ring
    • Chains
    • Dimension
    • Ideals
  • commutative ring
    • Noetherian
    • Rings
    • Mathematics
    • Isbn
    • Krull
    • Zero
    • Ring
    • Displaystyle
    • Prime
    • Chains
    • Dimension
    • Local
  • wolfgang krull
    • Ring
    • Field
    • Noetherian
    • Defined
    • Supremum
    • Zero
    • Possibly
    • Ideals
    • Lengths
    • Displaystyle
    • Prime
    • Rings
  • prime ideals
    • Prime
    • Height
    • Supremum
    • Ideal
    • Lengths
    • Displaystyle
    • Mathfrak
    • Chain
    • Maximal
    • Definition
    • Ring
    • Krull
  • noetherian ring
    • Noetherian
    • Ring
    • Rings
    • Displaystyle
    • Prime
    • Ideal
    • Local
    • Height
    • Zero
    • Example
    • Called
    • Defined
  • dimension of an algebraic variety
    • Geometry
    • Krull
    • Ring
    • Mathematics
    • Field
    • Noetherian
    • Zero
    • Rings
    • Defined
    • Definition
    • Supremum
    • Ideal

Connections between topic areas Semantic bridges

For Krull dimension, one of the stronger structural bridges in this analysis connects Krull dimension 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
Krull dimension — Overview · splits 57 ⟂ 20
Krull dimension — Examples · splits 60 ⟂ 17
Krull dimension — Bibliography · splits 60 ⟂ 17
Krull dimension — Explanation · splits 67 ⟂ 10
Krull dimension — Schemes · splits 71 ⟂ 6
Krull dimension — Of a module · splits 71 ⟂ 6

Map overview Semantic statistics

Krull dimension

Nodes77
Edges76
Triples40
Avg. degree1.97
Density0.025974
Components1

Source & methodology

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

Source: Wikipedia — Krull dimension · EN edition · Analysis: TopicsToTalkAbout

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