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13Exx Chain conditions and finite conditions

This subtopic studies chain conditions and finite conditions, focusing on Noetherian, Artinian, and related finiteness properties in rings and modules.

Specific topics

13E05 Commutative Noetherian rings and modules

Overview

13E05 develops commutative noetherian rings and modules in chain and finiteness conditions in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.

Related Wikipedia Page

Commutative Noetherian rings and modules (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and benchmark examples for commutative noetherian rings and modules
  • 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

Recommended Textbooks

13E10 Commutative Artinian rings and modules

Overview

13E10 develops commutative artinian rings and modules in chain and finiteness conditions in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.

Related Wikipedia Page

Commutative Artinian rings and modules (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and benchmark examples for commutative artinian rings and modules
  • 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

Recommended Textbooks

13E15 Commutative rings with finiteness conditions

Overview

13E15 develops commutative rings with finiteness conditions in chain and finiteness conditions in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.

Related Wikipedia Page

Commutative rings with finiteness conditions (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and benchmark examples for commutative rings with finiteness conditions
  • 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

Recommended Textbooks