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