Arf rings
E654173
Arf rings are a class of commutative rings introduced by Turkish mathematician Cahit Arf in his work on algebraic number theory and singularity theory, notable for their role in resolving certain types of singularities.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Arf rings canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7281889 — 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: Arf rings Context triple: [Cahit Arf, knownFor, Arf rings]
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A.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
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B.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
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C.
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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D.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
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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: Arf rings Target entity description: Arf rings are a class of commutative rings introduced by Turkish mathematician Cahit Arf in his work on algebraic number theory and singularity theory, notable for their role in resolving certain types of singularities.
-
A.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
-
B.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
C.
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.
-
D.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
-
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 (27)
| Predicate | Object |
|---|---|
| instanceOf |
class of commutative rings
ⓘ
mathematical concept ⓘ |
| appliesTo | one-dimensional analytically irreducible local domains ⓘ |
| characterizedBy |
conditions on value semigroup
ⓘ
stability under blowing up of ideals ⓘ |
| context |
one-dimensional local domains
ⓘ
valuation theory ⓘ |
| developedIn | 20th century ⓘ |
| field |
algebraic number theory
ⓘ
commutative algebra ⓘ singularity theory ⓘ |
| goal |
control of multiplicity sequences of singularities
ⓘ
simplification of resolution processes ⓘ |
| hasOrigin |
study of algebraic function fields
ⓘ
study of plane curve singularities ⓘ |
| influenced |
later work on numerical semigroups
ⓘ
later work on singularity resolution techniques ⓘ |
| introducedBy | Cahit Arf ⓘ |
| namedAfter | Cahit Arf ⓘ |
| property | commutative ⓘ |
| relatedTo |
Arf closure
ⓘ
Arf invariant ⓘ integral closure of rings ⓘ numerical semigroups ⓘ |
| usedIn |
algebraic geometry
ⓘ
resolution of singularities ⓘ study of curve singularities ⓘ |
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: Arf rings Description of subject: Arf rings are a class of commutative rings introduced by Turkish mathematician Cahit Arf in his work on algebraic number theory and singularity theory, notable for their role in resolving certain types of singularities.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.