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 , 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) Ideal theory 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 Dimension theory (algebra) 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

Official source

https://en.wikipedia.org/wiki/List_of_commutative_algebra_topics

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