Dedekind domain
E621104
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dedekind domain canonical | 3 |
| Dedekind domains | 1 |
| Dedekind ring | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833773 — 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: Dedekind domain Context triple: [Noetherian ring, isGeneralizationOf, Dedekind domain]
-
A.
Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
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B.
Henselian ring
A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.
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C.
ring theory
Ring theory is a branch of abstract algebra that studies rings—algebraic structures equipped with two binary operations generalizing addition and multiplication of integers—and their ideals, modules, and homomorphisms.
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D.
Krull dimension
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
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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: Dedekind domain Target entity description: 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.
-
A.
Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
-
B.
Henselian ring
A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.
-
C.
ring theory
Ring theory is a branch of abstract algebra that studies rings—algebraic structures equipped with two binary operations generalizing addition and multiplication of integers—and their ideals, modules, and homomorphisms.
-
D.
Krull dimension
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
-
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 (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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dedekind domain Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Noetherian ring
this entity surface form:
Dedekind ring
subject surface form:
Introduction to Commutative Algebra
this entity surface form:
Dedekind domains