• Source: 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




Z



{\displaystyle \mathbb {Z} }

, and p-adic integers.




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




Active research areas


Serre's multiplicity conjectures
homological conjectures




Related disciplines


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

Kata Kunci Pencarian:

• List of commutative algebra topics
• Commutative algebra
• List of abstract algebra topics
• List of homological algebra topics
• Lists of mathematics topics
• List of algebraic topology topics
• List of algebraic geometry topics
• Glossary of commutative algebra
• Operator algebra
• Homological algebra