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

Ideal theory

 Main article: Ideal theory

Dimension theory

 Main article: Dimension theory (algebra)