13G05 Integral domains

Overview

13G05 develops integral domains in local rings and related topics. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.

Related Wikipedia Page

Integral domains (Wikipedia)

Useful Links

• Integral domains (Wikipedia)
• Local ring (Wikipedia)
• Commutative algebra (Wikipedia)

Key Ideas

• Canonical formulations and benchmark examples for integral domains
• How local/global and categorical viewpoints interact in commutative settings
• Standard homological, finiteness, and multiplicative tools used in proofs

Typical Uses

Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.

Applications

• Commutative algebra foundations for algebraic geometry
• Algorithmic and symbolic computations in ring theory
• Interfaces with number theory, logic, and deformation methods

References

• Kaplansky, I. (1974) Commutative Rings. Rev. edn. Chicago, IL: University of Chicago Press.
• Nagata, M. (1962) Local Rings. New York, NY: Interscience.

Recommended Textbooks

• Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill.
• Topic-focused textbook search (n.d.) Integral domains. Available via Amazon search.