Triple

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).

Phrase: Artin’s refinement for Artinian rings | Statement: [Artin–Wedderburn theorem, hasComponentResult, Artin’s refinement for Artinian rings]

Target entity: Artin’s refinement for Artinian rings
Context triple: [Artin–Wedderburn theorem, hasComponentResult, Artin’s refinement for Artinian rings]

• A. 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.
• 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. Arf rings
  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.
• D. 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.
• E. 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.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Artin’s refinement for Artinian rings
Target entity description: Artin’s refinement for Artinian rings is a strengthened form of the Artin–Wedderburn theorem that characterizes the structure of Artinian rings more precisely, often via decompositions into matrix rings over division rings.

• A. 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.
• 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. Arf rings
  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.
• D. 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.
• E. 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.
• F. None of above. chosen