The Poincaré-Birkhoff-Witt theorem in ring theory

E239175

mathematical work research article

"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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All labels observed (4)

Label	Occurrences
Poincaré–Birkhoff–Witt theorem	4
The Poincaré-Birkhoff-Witt theorem in ring theory canonical	2
Poincaré–Birkhoff–Witt basis	1
Poincaré–Birkhoff–Witt-type basis (PBW-type)	1

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]

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
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	Poincaré–Birkhoff–Witt theorem ⓘ linked to: The Poincaré-Birkhoff-Witt theorem in ring theory
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	Poincaré–Birkhoff–Witt theorem ⓘ linked to: The Poincaré-Birkhoff-Witt theorem in ring theory
hasTitle	The Poincaré-Birkhoff-Witt theorem in ring theory ⓘ
language	English ⓘ
mainSubject	Lie algebras ⓘ Poincaré–Birkhoff–Witt theorem ⓘ linked to: The Poincaré-Birkhoff-Witt theorem in ring theory 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