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: Cohen–Macaulay ring | Statement: [Auslander–Buchsbaum formula, relatedConcept, Cohen–Macaulay ring]

Target entity: Cohen–Macaulay ring
Context triple: [Auslander–Buchsbaum formula, relatedConcept, Cohen–Macaulay ring]

• A. Castelnuovo–Mumford regularity
  Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
• B. 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.
• C. 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.
• 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–Gabriel dimension
  Krull–Gabriel dimension is a refinement of Krull dimension used in the representation theory of rings and abelian categories to measure the complexity of their subobject lattices and module categories.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"

Entity: Cohen–Macaulay ring
Triple: [Auslander–Buchsbaum formula, relatedConcept, Cohen–Macaulay ring]

A Cohen–Macaulay ring is a commutative Noetherian ring whose depth equals its Krull dimension, giving it especially well-behaved homological and geometric properties.

Target entity: Cohen–Macaulay ring
Target entity description: A Cohen–Macaulay ring is a commutative Noetherian ring whose depth equals its Krull dimension, giving it especially well-behaved homological and geometric properties.

• A. Castelnuovo–Mumford regularity
  Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
• B. 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.
• C. 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.
• 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–Gabriel dimension
  Krull–Gabriel dimension is a refinement of Krull dimension used in the representation theory of rings and abelian categories to measure the complexity of their subobject lattices and module categories.
• F. None of above. chosen