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