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.
, and Research fields
Active research areas
Basic notions
- Commutative ring
- Module (mathematics)
- Ring ideal, maximal ideal, prime ideal
- Ring homomorphism
- 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
Localization and completion
- Completion (ring theory)
- Formal power series
- Localization of a ring
- Regular local ring
- Localization of a module
- Valuation (mathematics)
- 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
Main article: Ideal theoryHomological 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
Main article: Dimension theory (algebra)- Height (ring theory)
- Depth (ring theory)
- Hilbert polynomial
- Regular local 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
Computational and algorithmic aspects
Active research areas
Related disciplines
- Algebraic number theory
- Algebraic geometry
- Ring theory
- Field theory (mathematics)
- Differential algebra
- Homological algebra
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