Concept

List of commutative algebra topics

Summary
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers \mathbb{Z}, and p-adic integers. Research fields
  • Combinatorial commutative algebra
  • Invariant theory
Active research areas
  • Serre's multiplicity conjectures
  • Homological conjectures
Basic notions
  • Commutative ring
  • Module (mathematics)
  • Ring ideal, maximal ideal, prime ideal
  • Ring homomorphism **Ring monomorphism **Ring epimorphism **Ring isomorphism
  • Zero divisor
  • Chinese remainder theorem
Classes of rings
  • Field (mathematics)
  • Algebraic number field
  • Polynomial ring
  • Integral domain
  • Boolean algebra (structure)
  • Principal ideal domain
  • Euclidean domain
  • Unique factorizati
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