Krull’s principal ideal theorem
E621107
Krull’s principal ideal theorem is a fundamental result in commutative algebra that relates the height of prime ideals containing a principal ideal to the Krull dimension of the ring.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Krull’s height theorem | 1 |
| Krull’s principal ideal theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833908 — 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’s principal ideal theorem Context triple: [Krull dimension, appearsIn, Krull’s principal ideal theorem]
-
A.
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.
-
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.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
D.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
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.
- 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’s principal ideal theorem Target entity description: Krull’s principal ideal theorem is a fundamental result in commutative algebra that relates the height of prime ideals containing a principal ideal to the Krull dimension of the ring.
-
A.
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.
-
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.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
D.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
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.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
result in ring theory
ⓘ
theorem in commutative algebra ⓘ |
| appearsIn |
Atiyah–Macdonald: Introduction to Commutative Algebra
NERFINISHED
ⓘ
Matsumura: Commutative Ring Theory NERFINISHED ⓘ standard textbooks on commutative algebra ⓘ |
| appliesTo | Noetherian rings NERFINISHED ⓘ |
| assumes |
Noetherian hypothesis on the ring
ⓘ
commutative ring with identity ⓘ |
| concerns |
Krull dimension
ⓘ
prime ideals ⓘ principal ideals ⓘ |
| consequence |
control on lengths of chains of prime ideals over principal ideals
ⓘ
dimension of a Noetherian domain is at least the transcendence degree of its field of fractions ⓘ |
| coreConcept |
Krull dimension of a ring
ⓘ
Noetherian condition NERFINISHED ⓘ height of a prime ideal ⓘ minimal prime over an ideal ⓘ |
| field |
algebraic geometry
ⓘ
commutative algebra ⓘ |
| formalizes | upper bound on codimension of varieties defined by one equation ⓘ |
| generalizationOf | dimension theory for polynomial rings ⓘ |
| givesBound | height of prime ideals minimal over a principal ideal ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies |
codimension of a hypersurface is at most 1
ⓘ
height of a principal prime ideal is at most 1 in a Noetherian ring ⓘ |
| namedAfter | Wolfgang Krull NERFINISHED ⓘ |
| proofTechniques |
Noether normalization
NERFINISHED
ⓘ
induction on dimension ⓘ localization of rings ⓘ |
| relatedTo |
Krull’s dimension theory
NERFINISHED
ⓘ
Krull’s height theorem NERFINISHED ⓘ Krull’s principal ideal theorem for finitely generated ideals ⓘ |
| relates |
dimension of a ring
ⓘ
height of prime ideals ⓘ number of generators of an ideal ⓘ |
| statementForm | inequality on heights of prime ideals ⓘ |
| typicalContext |
affine algebras over a field
ⓘ
local rings ⓘ |
| typicalFormulation | If R is Noetherian and x in R, then every minimal prime over (x) has height at most 1 ⓘ |
| usedIn |
algebraic geometry via coordinate rings
ⓘ
dimension theory of Noetherian rings ⓘ study of chains of prime ideals ⓘ theory of regular local rings ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Krull’s principal ideal theorem Description of subject: Krull’s principal ideal theorem is a fundamental result in commutative algebra that relates the height of prime ideals containing a principal ideal to the Krull dimension of the ring.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Krull’s height theorem