13Axx General commutative algebra
This subtopic studies general commutative algebra, covering rings, ideals, modules, and the fundamental structural ideas used throughout the subject.
Specific topics
13A02 Graded rings
Overview
13A02 develops graded rings within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Graded rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for graded rings
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A05 Divisibility; factorizations
Overview
13A05 develops divisibility; factorizations within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Divisibility; factorizations (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for divisibility; factorizations
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A15 Ideals and multiplicative ideal theory
Overview
13A15 develops ideals and multiplicative ideal theory within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Ideals and multiplicative ideal theory (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for ideals and multiplicative ideal theory
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A18 Valuations and their generalizations
Overview
13A18 develops valuations and their generalizations within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Valuations and their generalizations (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for valuations and their generalizations
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A30 Associated graded rings of ideals; Rees ring; blowing-up
Overview
13A30 develops associated graded rings of ideals; rees ring; blowing-up within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Associated graded rings of ideals; Rees ring; blowing-up (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for associated graded rings of ideals; rees ring; blowing-up
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A35 Characteristic $p$ methods; tight closure
Overview
13A35 develops characteristic p methods; tight closure within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Characteristic p methods; tight closure (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for characteristic p methods; tight closure
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A50 Actions of groups on commutative rings
Overview
13A50 develops actions of groups on commutative rings within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Actions of groups on commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for actions of groups on commutative rings
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks
13A99 None of the above
Overview
13A99 develops none of the above within general commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
None of the above (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for none of the above
- Interactions between ring/field structure, morphisms, and module behavior
- Common proof patterns and invariants used in modern algebraic practice
Typical Uses
Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.
Applications
- Core theoretical developments in algebra and arithmetic geometry
- Symbolic computation and algorithmic algebra workflows
- Foundational support for logic, coding theory, and number theory links
References
Recommended Textbooks