List of commutative algebra topics

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 factorization domain
• Dedekind domain
• Nilpotent elements and reduced rings
• Dual numbers
• Tensor product of fields
• Tensor product of R-algebras

Constructions with commutative rings

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

Localization and completion

• 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

Finiteness properties

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

Ideal theory

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

Homological properties

• 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

Dimension theory

• 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

Ring extensions, primary decomposition

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

Relation with algebraic geometry

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

Computational and algorithmic aspects

• Elimination theory
• Gröbner basis
• Buchberger's algorithm

Related disciplines

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