Commutative algebra , first known as ideal theory, is the branch of 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 ${\displaystyle \mathbb {Z} }$; and p-adic integers.

Contents

• Research fields
• Active research areas
• Basic notions
• Classes of rings
• Constructions with commutative rings
• Localization and completion
• Finiteness properties
• Ideal theory
• Homological properties
• Dimension theory
• Ring extensions, primary decomposition
• Relation with algebraic geometry
• Computational and algorithmic aspects
• Related disciplines

Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts.

Research fields

• Combinatorial commutative algebra
• Invariant theory

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