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.

Aliases (3)

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 → 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	finiteness theorem →
usedIn	algorithmic ideal theory in polynomial rings → proofs of Hilbert's Nullstellensatz → proofs of Noether normalization lemma →
usesConcept	Noetherian ring → ascending chain condition → finitely generated ideal → ideal → polynomial ring →

Referenced by (7)

Subject (surface form when different)	Predicate
Hilbert’s Nullstellensatz ("Hilbert’s basis theorem") → Hilbert’s syzygy theorem ("Hilbert’s basis theorem") → Noether normalization lemma → Noetherian module →	relatedTo
David Hilbert → Grete Hermann ("Die Frage der endlich vielen Schritte in der Theorie der Polynomideale") →	notableWork
Hilbert basis theorem ("finiteness theorem") →	typeOf