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