Dedekind domain

E621104

integral domain ring-theoretic structure

A Dedekind domain is an integral domain in which every nonzero proper ideal factors uniquely into a product of prime ideals, playing a central role in algebraic number theory and the study of rings of integers in number fields.

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Observed surface forms (2)

Surface form	Occurrences
Dedekind domains	1
Dedekind ring	1

Statements (48)

Predicate	Object
instanceOf	integral domain ⓘ ring-theoretic structure ⓘ
centralRoleIn	ideal-theoretic approach to algebraic number theory ⓘ theory of rings of integers in number fields ⓘ
closureProperty	finite integral extensions of Dedekind domains are Dedekind domains (under suitable hypotheses) ⓘ localization at multiplicative sets yields Dedekind domains (under suitable conditions) ⓘ
equivalentCondition	Noetherian + integrally closed + Krull dimension 1 ⓘ every localization at a nonzero prime ideal is a discrete valuation ring ⓘ every nonzero fractional ideal is invertible ⓘ every nonzero ideal factors uniquely as a product of prime ideals ⓘ
field	algebraic number theory ⓘ commutative algebra ⓘ
generalizes	principal ideal domain ⓘ unique factorization domain via ideals ⓘ
hasExample	coordinate ring of a smooth projective curve over a field minus a point ⓘ discrete valuation ring ⓘ principal ideal domain ⓘ ring of integers Z of the rational numbers ⓘ ring of integers of a number field ⓘ ring of integers of a quadratic number field ⓘ
hasInvariant	discriminant of its field of fractions (in number field case) ⓘ ideal class group ⓘ unit group ⓘ
hasProperty	Krull dimension 1 ⓘ Noetherian ⓘ every nonzero fractional ideal is invertible ⓘ every nonzero ideal can be written as a finite product of prime ideals ⓘ every nonzero ideal is invertible ⓘ every nonzero proper ideal factors uniquely into prime ideals ⓘ integrally closed in its field of fractions ⓘ locally a discrete valuation ring at every nonzero prime ideal ⓘ one-dimensional Noetherian normal domain ⓘ unique factorization of ideals ⓘ
implies	class group is defined as group of fractional ideals modulo principal ideals ⓘ every nonzero ideal has a unique factorization into powers of distinct prime ideals ⓘ every nonzero prime ideal is maximal ⓘ
namedAfter	Richard Dedekind NERFINISHED ⓘ
nonExample	non-Noetherian integrally closed domain of dimension 1 ⓘ polynomial ring in two variables over a field ⓘ
relatedConcept	Krull domain NERFINISHED ⓘ Noetherian domain NERFINISHED ⓘ discrete valuation ring ⓘ integrally closed domain ⓘ principal ideal domain ⓘ unique factorization domain ⓘ
usedFor	defining and studying ideal class groups ⓘ studying arithmetic of number fields ⓘ studying failure of unique factorization of elements via ideals ⓘ

Referenced by (5)

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

Noetherian rings → isGeneralizationOf → Dedekind domain ⓘ

subject surface form: Noetherian ring

Richard Dedekind → knownFor → Dedekind domain ⓘ

Richard Dedekind → hasConceptNamedAfter → Dedekind domain ⓘ

this entity surface form: Dedekind ring

“Introduction to Commutative Algebra” (with Ian G. Macdonald) → hasSubject → Dedekind domain ⓘ

subject surface form: Introduction to Commutative Algebra

this entity surface form: Dedekind domains