13Cxx Theory of modules and ideals
This subtopic studies the theory of modules and ideals, analyzing decomposition, homological properties, and the interplay between ring structure and module behavior.
Specific topics
13C05 Structure, classification theorems for modules and ideals
Overview
13C05 develops structure, classification theorems for modules and ideals within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Structure, classification theorems for modules and ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for structure, classification theorems for modules and ideals
- 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
13C10 Projective and free modules and ideals
Overview
13C10 develops projective and free modules and ideals within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Projective and free modules and ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for projective and free modules and ideals
- 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
13C11 Injective and flat modules and ideals
Overview
13C11 develops injective and flat modules and ideals within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Injective and flat modules and ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for injective and flat modules and ideals
- 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
13C12 Torsion modules and ideals
Overview
13C12 develops torsion modules and ideals within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Torsion modules and ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for torsion modules and ideals
- 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
13C13 Other special types of modules and ideals
Overview
13C13 develops other special types of modules and ideals within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Other special types of modules and ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for other special types of modules and ideals
- 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
13C14 Cohen-Macaulay modules
Overview
13C14 develops cohen-macaulay modules within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Cohen-Macaulay modules (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for cohen-macaulay modules
- 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
13C15 Dimension theory, depth, related commutative rings
Overview
13C15 develops dimension theory, depth, related commutative rings within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Dimension theory, depth, related commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for dimension theory, depth, related 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
13C20 Class groups
Overview
13C20 develops class groups within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Class groups (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for class groups
- 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
13C40 Linkage, complete intersections and determinantal ideals
Overview
13C40 develops linkage, complete intersections and determinantal ideals within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Linkage, complete intersections and determinantal ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for linkage, complete intersections and determinantal ideals
- 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
13C60 Module categories and commutative rings
Overview
13C60 develops module categories and commutative rings within modules and ideals in commutative algebra. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Module categories and commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for module categories and 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