List of commutative algebra topics
Research fields
Active research areas
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
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 and descending chain condition
Ideal theory
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