Triple

T9839070
Position Surface form Disambiguated ID Type / Status
Subject The Poincaré-Birkhoff-Witt theorem in ring theory E239175 entity
Predicate hasTitle P38 FINISHED
Object The Poincaré-Birkhoff-Witt theorem in ring theory E239175 NE FINISHED

How this triple was built (2 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: The Poincaré-Birkhoff-Witt theorem in ring theory | Statement: [The Poincaré-Birkhoff-Witt theorem in ring theory, hasTitle, The Poincaré-Birkhoff-Witt theorem in ring theory]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: The Poincaré-Birkhoff-Witt theorem in ring theory
Context triple: [The Poincaré-Birkhoff-Witt theorem in ring theory, hasTitle, The Poincaré-Birkhoff-Witt theorem in ring theory]
  • A. The Poincaré-Birkhoff-Witt theorem in ring theory chosen
    "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. Lasker–Noether theorem on primary decomposition
    The Lasker–Noether theorem on primary decomposition is a fundamental result in commutative algebra stating that every ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals, generalizing the factorization of integers into prime powers.
  • C. Gelfand–Kirillov dimension
    The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
  • D. 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.
  • 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.
  • G. Unsure - the case is ambiguous/there is not enough information to decide.

Provenance (3 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ca84e3f0c48190ada72a65ebd50efd completed March 30, 2026, 2:12 p.m.
NER Named-entity recognition batch_69cdb34921b881909836ba0f5b42a27b completed April 2, 2026, 12:07 a.m.
NED1 Entity disambiguation (via context triple) batch_69d20d4079048190976eb198f8ef62f0 completed April 5, 2026, 7:20 a.m.
Created at: March 30, 2026, 8:33 p.m.