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Concept# List of commutative algebra topics

Summary

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

- Combinatorial commutative algebra
- Invariant theory

- Serre's multiplicity conjectures
- Homological conjectures

- Commutative ring
- Module (mathematics)
- Ring ideal, maximal ideal, prime ideal
- Ring homomorphism **Ring monomorphism **Ring epimorphism **Ring isomorphism
- Zero divisor
- Chinese remainder theorem

- Field (mathematics)
- Algebraic number field
- Polynomial ring
- Integral domain
- Boolean algebra (structure)
- Principal ideal domain
- Euclidean domain
- Unique factorizati

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