The Poincaré-Birkhoff-Witt theorem in ring theory
E239175
"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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Poincaré–Birkhoff–Witt theorem | 3 |
| The Poincaré-Birkhoff-Witt theorem in ring theory canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2169655 — 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: The Poincaré-Birkhoff-Witt theorem in ring theory Context triple: [N. G. de Bruijn, hasPublication, The Poincaré-Birkhoff-Witt theorem in ring theory]
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A.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
B.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
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C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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D.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
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E.
Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth 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: The Poincaré-Birkhoff-Witt theorem in ring theory Target entity description: "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.
-
A.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
B.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
-
E.
Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
- F. None of above. chosen
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical work
ⓘ
research article ⓘ |
| about |
basis properties derived from the Poincaré–Birkhoff–Witt theorem
ⓘ
embedding of Lie algebras into associative algebras ⓘ structure of enveloping algebras as modules ⓘ |
| applies |
The Poincaré-Birkhoff-Witt theorem in ring theory
self-linksurface differs
ⓘ
surface form:
Poincaré–Birkhoff–Witt theorem
|
| author | N. G. de Bruijn ⓘ |
| concerns |
Lie rings
ⓘ
algebraic structures over a ring ⓘ associative rings ⓘ |
| field |
Lie theory
ⓘ
algebra ⓘ ring theory ⓘ |
| focusesOn |
basis theorems for enveloping algebras
ⓘ
relations between Lie algebras and associative algebras ⓘ universal enveloping algebras ⓘ |
| hasTheoremAsTopic |
The Poincaré-Birkhoff-Witt theorem in ring theory
self-linksurface differs
ⓘ
surface form:
Poincaré–Birkhoff–Witt theorem
|
| hasTitle | The Poincaré-Birkhoff-Witt theorem in ring theory self-link ⓘ |
| language | English ⓘ |
| mainSubject |
Lie algebras
ⓘ
The Poincaré-Birkhoff-Witt theorem in ring theory self-linksurface differs ⓘ
surface form:
Poincaré–Birkhoff–Witt theorem
ring theory ⓘ |
| studies |
Lie-theoretic ring structures
ⓘ
associative ring structures ⓘ |
| usesConcept |
Lie algebra
ⓘ
algebra homomorphism ⓘ associative algebra ⓘ graded structures ⓘ ring ⓘ universal enveloping algebra ⓘ |
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: The Poincaré-Birkhoff-Witt theorem in ring theory Description of subject: "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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
The Poincaré-Birkhoff-Witt theorem in ring theory
→
mainSubject
→
The Poincaré-Birkhoff-Witt theorem in ring theory
self-linksurface differs
ⓘ
this entity surface form:
Poincaré–Birkhoff–Witt theorem
The Poincaré-Birkhoff-Witt theorem in ring theory
→
applies
→
The Poincaré-Birkhoff-Witt theorem in ring theory
self-linksurface differs
ⓘ
this entity surface form:
Poincaré–Birkhoff–Witt theorem
The Poincaré-Birkhoff-Witt theorem in ring theory
→
hasTheoremAsTopic
→
The Poincaré-Birkhoff-Witt theorem in ring theory
self-linksurface differs
ⓘ
this entity surface form:
Poincaré–Birkhoff–Witt theorem
The Poincaré-Birkhoff-Witt theorem in ring theory
→
hasTitle
→
The Poincaré-Birkhoff-Witt theorem in ring theory
self-link
ⓘ