Chinese remainder theorem in commutative algebra

Target entity: Chinese remainder theorem in commutative algebra
Context triple: [Chinese remainder theorem, hasGeneralization, Chinese remainder theorem in commutative algebra]

• A. 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.
• B. 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.
• C. Eisenbud’s Commutative Algebra
  Eisenbud’s *Commutative Algebra* is a widely used graduate-level textbook that develops modern commutative algebra with strong connections to algebraic geometry, featuring topics such as free resolutions, syzygies, and Castelnuovo–Mumford regularity.
• D. 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.
• E. Cours d’algèbre commutative
  Cours d’algèbre commutative is a foundational French textbook on commutative algebra authored by mathematician Jean-Daniel Perrin.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Chinese remainder theorem in commutative algebra
Target entity description: The Chinese remainder theorem in commutative algebra is a generalization of the classical Chinese remainder theorem that characterizes how a commutative ring decomposes via its ideals, typically expressing a ring as a product of quotient rings under suitable comaximality conditions.

• A. 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.
• B. 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.
• C. Eisenbud’s Commutative Algebra
  Eisenbud’s *Commutative Algebra* is a widely used graduate-level textbook that develops modern commutative algebra with strong connections to algebraic geometry, featuring topics such as free resolutions, syzygies, and Castelnuovo–Mumford regularity.
• D. 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.
• E. Cours d’algèbre commutative
  Cours d’algèbre commutative is a foundational French textbook on commutative algebra authored by mathematician Jean-Daniel Perrin.
• F. None of above. chosen