Wikipedia:
List of commutative algebra topics
This is a list of commutative algebra topics, by Wikipedia page.
Research fields
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 , reduced ring- dual numbers
- tensor product of fields
- tensor product of R-algebras
Constructions with commutative rings
Localization and completion
- completion (ring theory)
- formal power series
- local ring
- regular local ring
localization of a 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
Homological properties
- flat module
- flat map
- 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
- global dimension
- regular sequence (algebra)
- Krull dimension
- Krull's principal ideal theorem
Ring extensions, primary decomposition
- primary ideal
- primary decomposition, Lasker–Noether theorem
- Noether normalization lemma
- going up and going down
Relation with algebraic geometry
Computational and algorithmic aspects
Active research areas
Related disciplines
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