Dedekind ideal
E634837
A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dedekind ideal canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7011038 — 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 ideal Context triple: [Richard Dedekind, knownFor, Dedekind ideal]
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A.
Dedekind domain
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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B.
Jacobson radical
The Jacobson radical is an ideal of a ring that captures elements annihilating all simple modules, playing a key role in understanding the ring’s structure and its representations.
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C.
Fitting ideal
The Fitting ideal is an algebraic invariant in commutative algebra and module theory that encodes information about the structure and relations of a finitely generated module over a ring.
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D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
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E.
Krull’s principal ideal theorem
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.
- 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 ideal Target entity description: A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
-
A.
Dedekind domain
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.
-
B.
Jacobson radical
The Jacobson radical is an ideal of a ring that captures elements annihilating all simple modules, playing a key role in understanding the ring’s structure and its representations.
-
C.
Fitting ideal
The Fitting ideal is an algebraic invariant in commutative algebra and module theory that encodes information about the structure and relations of a finitely generated module over a ring.
-
D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
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E.
Krull’s principal ideal theorem
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.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
ideal in ring theory
ⓘ
mathematical concept ⓘ |
| appearsIn |
algebraic number theory textbooks
ⓘ
commutative algebra textbooks ⓘ |
| field |
algebraic number theory
ⓘ
ring theory ⓘ |
| hasProperty |
every nonzero proper ideal factors uniquely into prime ideals in a Dedekind domain
ⓘ
nonzero ideals correspond to finitely generated torsion modules over a Dedekind domain ⓘ nonzero ideals in a Dedekind domain are invertible fractional ideals ⓘ nonzero ideals in a Dedekind domain have unique factorization up to order ⓘ nonzero prime ideals in a Dedekind domain are maximal ⓘ |
| isDefinedIn | commutative ring with identity ⓘ |
| isNamedAfter | Richard Dedekind ⓘ |
| isRelatedTo |
Dedekind domain
ⓘ
Krull dimension one ⓘ Noetherian ring ⓘ
linked to:
Noetherian rings
class number ⓘ fractional ideal ⓘ fractional ideal group ⓘ ideal class group ⓘ ideal factorization ⓘ ideal inversion ⓘ ideal multiplication ⓘ ideal norm ⓘ integral ideal ⓘ integrally closed domain ⓘ localization of rings ⓘ maximal ideal ⓘ prime decomposition in number fields ⓘ prime ideal ⓘ principal ideal ⓘ principal ideal domain ⓘ unique factorization domain ⓘ unique factorization of ideals ⓘ valuation theory ⓘ |
| isStudiedIn | Dedekind domain ⓘ |
| isUsedFor |
decomposing prime ideals in extensions of number fields
ⓘ
defining ideal class group ⓘ studying arithmetic of algebraic number fields ⓘ studying failure of unique factorization of elements ⓘ studying ramification in number fields ⓘ studying splitting of primes in extensions ⓘ |
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 ideal Description of subject: A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.