Krull’s principal ideal theorem

E621107

result in ring theory theorem in commutative algebra

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.

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All labels observed (3)

Label	Occurrences
Krull’s principal ideal theorem canonical	2
Krull’s height theorem	1
principal ideal theorem	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 ⓘ linked to: “Introduction to Commutative Algebra” (with Ian G. Macdonald) Matsumura: Commutative Ring Theory ⓘ standard textbooks on commutative algebra ⓘ
appliesTo	Noetherian rings ⓘ
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 ⓘ linked to: Noetherian rings 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 ⓘ
proofTechniques	Noether normalization ⓘ linked to: Noether normalization lemma induction on dimension ⓘ localization of rings ⓘ
relatedTo	Krull’s dimension theory ⓘ linked to: Krull dimension Krull’s height theorem ⓘ linked to: Krull dimension 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

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Krull dimension → appearsIn → Krull’s principal ideal theorem ⓘ

Krull dimension → appearsIn → Krull’s height theorem ⓘ

linked to: Krull’s principal ideal theorem

Lasker–Noether theorem on primary decomposition → relatedTo → Krull’s principal ideal theorem ⓘ

Furtwängler’s theorem in class field theory → refines → principal ideal theorem ⓘ