List of commutative algebra topics

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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 \mathbb{Z}, and p-adic integers.

Research fields[edit]

Active research areas[edit]

Basic notions[edit]

Classes of rings[edit]

Constructions with commutative rings[edit]

Localization and completion[edit]

Finiteness properties[edit]

Ideal theory[edit]

Main article: Ideal theory

Homological properties[edit]

Dimension theory[edit]

Ring extensions, primary decomposition[edit]

Relation with algebraic geometry[edit]

Computational and algorithmic aspects[edit]

Active research areas[edit]

Related disciplines[edit]