Hilbert basis theorem

E41778

mathematical theorem theorem in commutative algebra

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.

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All labels observed (5)

Label	Occurrences
Hilbert basis theorem canonical	3
Hilbert’s basis theorem	3
Die Frage der endlich vielen Schritte in der Theorie der Polynomideale	1
Hilbert basis theorem: if R is Noetherian then R[x] is Noetherian	1
finiteness theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T326972 — 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: Hilbert basis theorem
Context triple: [David Hilbert, notableWork, Hilbert basis theorem]

A. 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.
B. Noetherian induction
Noetherian induction is a proof technique used in mathematics to establish statements about structures satisfying the descending chain condition, generalizing ordinary mathematical induction.
C. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
D. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
E. Noetherian module
A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
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: Hilbert basis theorem
Target entity description: 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.

A. 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.
B. Noetherian induction
Noetherian induction is a proof technique used in mathematics to establish statements about structures satisfying the descending chain condition, generalizing ordinary mathematical induction.
C. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
D. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
E. Noetherian module
A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in commutative algebra ⓘ
appearsIn	standard graduate textbooks on algebraic geometry ⓘ standard graduate textbooks on commutative algebra ⓘ
assumes	commutative ring with identity ⓘ
characterizes	stability of Noetherian property under finite polynomial extension ⓘ
doesNotGenerallyHoldFor	polynomial rings in infinitely many variables ⓘ
field	algebraic geometry ⓘ commutative algebra ⓘ
formalizes	finite generation of ideals in polynomial rings over Noetherian rings ⓘ
generalizationOf	finiteness of ideals in principal ideal domains ⓘ
hasConsequence	Noetherian property is stable under adjoining finitely many polynomial variables ⓘ basis for the theory of Gröbner bases ⓘ every ideal in a polynomial ring over a field has a finite generating set ⓘ existence of finite generating sets for ideals in coordinate rings of affine varieties ⓘ finiteness properties in algebraic geometry ⓘ
historicalPeriod	late 19th century ⓘ
holdsFor	R[x] when R is Noetherian ⓘ R[x_1,\dots,x_n] when R is Noetherian ⓘ
implies	ascending chain condition on ideals holds in polynomial rings over Noetherian rings ⓘ coordinate rings of affine varieties over a field are Noetherian ⓘ every ideal in R[x_1,\dots,x_n] is finitely generated when R is Noetherian ⓘ every ideal in k[x_1,\dots,x_n] is finitely generated for any field k ⓘ k[x_1,\dots,x_n] is Noetherian for any field k ⓘ polynomial rings over Noetherian rings are Noetherian ⓘ
introducedInContextOf	Hilbert's work on invariant theory ⓘ
isFoundationFor	computational algebraic geometry ⓘ modern commutative algebra ⓘ
namedAfter	David Hilbert ⓘ
proofTechnique	induction on the number of variables ⓘ use of leading coefficients and degrees of polynomials ⓘ
relatedTo	Gröbner basis ⓘ Hilbert’s Nullstellensatz ⓘ surface form: Hilbert's Nullstellensatz Noether normalization lemma ⓘ Noetherian module ⓘ
states	If R is a Noetherian ring then the polynomial ring R[x] is Noetherian ⓘ If R is a Noetherian ring then the polynomial ring R[x_1,\dots,x_n] is Noetherian for any finite n ⓘ
typeOf	Hilbert basis theorem self-linksurface differs ⓘ surface form: finiteness theorem
usedIn	algorithmic ideal theory in polynomial rings ⓘ proofs of Hilbert's Nullstellensatz ⓘ proofs of Noether normalization lemma ⓘ
usesConcept	Noetherian rings ⓘ surface form: Noetherian ring ascending chain condition ⓘ finitely generated ideal ⓘ ideal ⓘ polynomial ring ⓘ

How these facts were elicited

Referenced by (9)

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

David Hilbert → notableWork → Hilbert basis theorem ⓘ

Noetherian module → relatedTo → Hilbert basis theorem ⓘ

Noether normalization lemma → relatedTo → Hilbert basis theorem ⓘ

Grete Hermann → notableWork → Hilbert basis theorem ⓘ

this entity surface form: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale

Hilbert basis theorem → typeOf → Hilbert basis theorem self-linksurface differs ⓘ

this entity surface form: finiteness theorem

Hilbert’s syzygy theorem → relatedTo → Hilbert basis theorem ⓘ

this entity surface form: Hilbert’s basis theorem

Hilbert’s Nullstellensatz → relatedTo → Hilbert basis theorem ⓘ

this entity surface form: Hilbert’s basis theorem

Noetherian rings → hasTheorem → Hilbert basis theorem ⓘ

subject surface form: Noetherian ring

this entity surface form: Hilbert basis theorem: if R is Noetherian then R[x] is Noetherian

Hilbert’s fourteenth problem → relatedTo → Hilbert basis theorem ⓘ

this entity surface form: Hilbert’s basis theorem

All labels observed (5)

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Statements (46)

How these facts were elicited Show

Referenced by (9)

How this entity was disambiguated

How these facts were elicited