List of commutative algebra topics

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 ${\displaystyle \mathbb{\mathbb{Z}}}$, and p-adic integers.

• Combinatorial commutative algebra
• Invariant theory

• Serre's multiplicity conjectures
• Homological conjectures

• Commutative ring
• Module (mathematics)
• Ring ideal, maximal ideal, prime ideal
• Ring homomorphism
  • Ring monomorphism
  • Ring epimorphism
  • Ring isomorphism
• Zero divisor
• Chinese remainder theorem

• Field (mathematics)
• Algebraic number field
• Polynomial ring
• Integral domain
• Boolean algebra (structure)
• Principal ideal domain
• Euclidean domain
• Unique factorization domain
• Dedekind domain
• Nilpotent elements and reduced rings
• Dual numbers
• Tensor product of fields
• Tensor product of R-algebras

• Quotient ring
• Field of fractions
• Product of rings
• Annihilator (ring theory)
• Integral closure

• Completion (ring theory)
• Formal power series
• Localization of a ring
  • Local ring
• Regular local ring
• Localization of a module
• Valuation (mathematics)
  • Discrete valuation
  • Discrete valuation ring
• I-adic topology
• Weierstrass preparation theorem

• Noetherian ring
• Hilbert's basis theorem
• Artinian ring
• Ascending chain condition (ACC) and descending chain condition (DCC)

• Fractional ideal
• Ideal class group
• Radical of an ideal
• Hilbert's Nullstellensatz

• Flat module
• Flat map
• Flat map (ring theory)
• Projective module
• Injective module
• Cohen-Macaulay ring
• Gorenstein ring
• Complete intersection ring
• Koszul complex
• Hilbert's syzygy theorem
• Quillen–Suslin theorem

• Height (ring theory)
• Depth (ring theory)
• Hilbert polynomial
• Regular local ring
  • Discrete valuation ring
• Global dimension
• Regular sequence (algebra)
• Krull dimension
• Krull's principal ideal theorem

• Primary ideal
• Primary decomposition and the Lasker–Noether theorem
• Noether normalization lemma
• Going up and going down

• Spectrum of a ring
• Zariski tangent space
• Kähler differential

• Elimination theory
• Gröbner basis
• Buchberger's algorithm

• Serre's multiplicity conjectures
• homological conjectures

• Algebraic number theory
• Algebraic geometry
• Ring theory
• Field theory (mathematics)
• Differential algebra
• Homological algebra

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