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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Observed surface forms (1)

Surface form	Occurrences
Krull’s height theorem	1

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 ⓘ

Referenced by (2)

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

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

this entity surface form: Krull’s height theorem