Henselian ring
E483407
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Henselian ring canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T4962223 — 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: Henselian ring Context triple: [Kurt Hensel, developedConcept, Henselian ring]
-
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.
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.
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.
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D.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
E.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
- 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: Henselian ring Target entity description: 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.
-
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.
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.
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.
-
D.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
E.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure property
ⓘ
ring theory concept ⓘ |
| allows |
lifting of certain polynomial factorizations from residue field to the ring
ⓘ
lifting of simple roots from residue field to the ring ⓘ |
| appearsIn |
study of local behavior of morphisms of schemes
ⓘ
theory of lifting solutions of polynomial equations ⓘ |
| contrastsWith | non-Henselian local ring ⓘ |
| field | commutative algebra ⓘ |
| generalizes | complete Noetherian local ring ⓘ |
| hasComponent |
maximal ideal
ⓘ
residue field ⓘ |
| hasConstruction | henselization of a given local ring ⓘ |
| hasDefinition | a local ring in which Hensel’s lemma holds ⓘ |
| hasEquivalentCondition |
every finite ring extension decomposes according to its residue field decomposition
ⓘ
idempotents of any finite algebra over the ring lift from the residue algebra ⓘ |
| hasExample |
Henselization of a local ring
ⓘ
complete local ring with respect to its maximal ideal ⓘ p-adic integers Z_p ⓘ strictly Henselian local ring ⓘ |
| hasProperty |
Henselian property is invariant under isomorphism of local rings
ⓘ
Henselian property is local on Spec in the sense of local rings at points ⓘ Henselian property is preserved under integral closure in finite extensions of fraction fields (under suitable hypotheses) ⓘ Henselian property is stable under finite products of local rings ⓘ Hensel’s lemma holds for monic polynomials ⓘ completion of a Noetherian local ring is Henselian ⓘ if a local ring is strictly Henselian then it is Henselian ⓘ if the residue field is separably closed and the ring is Henselian, then the ring is strictly Henselian ⓘ local ring ⓘ maximal ideal is contained in the Jacobson radical ⓘ |
| hasSubClass |
complete local ring
ⓘ
strictly Henselian ring ⓘ |
| hasUniversalProperty | henselization is initial among Henselian local rings receiving a local homomorphism from the given ring ⓘ |
| implies | uniqueness of lifting of simple roots under suitable conditions ⓘ |
| isCharacterizedBy |
every finite algebra over the ring decomposes according to its residue algebra decomposition
ⓘ
factorizations of monic polynomials over the residue field lift to factorizations over the ring under suitable conditions ⓘ idempotents lift uniquely from residue ring to the ring ⓘ |
| isNotEquivalentTo | complete local ring in general ⓘ |
| namedAfter | Kurt Hensel NERFINISHED ⓘ |
| relatedTo |
Henselization
ⓘ
henselization functor ⓘ étale morphism ⓘ |
| satisfies | Hensel’s lemma NERFINISHED ⓘ |
| usedIn |
algebraic geometry
ⓘ
local study of schemes ⓘ number theory ⓘ valuation theory ⓘ |
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: Henselian ring Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.