# List of commutative algebra topics

Jump to navigation
Jump to search

**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.

## 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

{{#invoke:main|main}}

## 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

{{#invoke:main|main}}

- 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