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

Classes of rings

Constructions with commutative rings

Localization and completion

Finiteness properties

Ideal theory

Main article: Ideal theory

Homological properties

Dimension theory

Main article: Dimension theory (algebra)

Ring extensions, primary decomposition

Relation with algebraic geometry

Computational and algorithmic aspects

Active research areas

Related disciplines