Krull dimension
E157401
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Krull dimension canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382890 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Krull dimension Context triple: [Noether normalization lemma, relatedTo, Krull dimension]
-
A.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
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C.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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E.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Krull dimension Target entity description: Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
-
A.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
C.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
commutative algebra concept
ⓘ
dimension theory concept ⓘ ring invariant ⓘ |
| appearsIn |
Hilbert’s Nullstellensatz context
ⓘ
Krull’s principal ideal theorem ⓘ
surface form:
Krull’s height theorem
Krull’s principal ideal theorem ⓘ |
| appliesTo |
Noetherian rings
ⓘ
surface form:
Noetherian ring
commutative ring with identity ⓘ integral domain ⓘ local ring ⓘ scheme ⓘ |
| comparedTo |
global dimension
ⓘ
homological dimension ⓘ projective dimension ⓘ |
| context |
used to measure complexity of algebraic sets over a ring
ⓘ
used to study chains of prime ideals in Spec R ⓘ |
| definedAs | the supremum of lengths of chains of prime ideals in a ring ⓘ |
| defines | a notion of dimension for commutative rings ⓘ |
| example |
a field has Krull dimension 0
ⓘ
a principal ideal domain that is not a field has Krull dimension 1 ⓘ an Artinian ring has Krull dimension 0 ⓘ the polynomial ring k[x1,…,xn] over a field k has Krull dimension n ⓘ |
| field |
algebraic geometry
ⓘ
commutative algebra ⓘ ring theory ⓘ |
| generalizes | geometric dimension of affine varieties ⓘ |
| hasVariant |
Krull–Gabriel dimension
ⓘ
cohomological dimension ⓘ transcendence degree as a related notion for fields ⓘ |
| measures | the size of a ring via chains of prime ideals ⓘ |
| namedAfter | Wolfgang Krull ⓘ |
| property |
can be infinite
ⓘ
equals the supremum of heights of prime ideals ⓘ for Noetherian rings equals the maximum length of chains of prime ideals ⓘ is a nonnegative integer or infinity ⓘ is invariant under ring isomorphism ⓘ is monotone under integral extensions in many cases ⓘ |
| relatedTo |
Zariski topology
ⓘ
codimension ⓘ dimension of an algebraic variety ⓘ height of a prime ideal ⓘ spectrum of a ring ⓘ |
| symbol | dim R ⓘ |
| usedIn |
algebraic geometry over Spec R
ⓘ
classification of Noetherian rings ⓘ commutative algebra textbooks ⓘ dimension theory of schemes ⓘ |
| usesConcept |
chain of prime ideals
ⓘ
prime ideal ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Krull dimension Description of subject: Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.