13Bxx Ring extensions and related structures
This subtopic studies ring extensions and related structures, including integral extensions, normalization, and the behavior of algebraic objects under extension.
Specific topics
13B02 Extension theory
Overview
13B02 develops extension theory within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Extension theory (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for extension 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
13B05 Galois theory and commutative ring extensions
Overview
13B05 develops galois theory and commutative ring extensions within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Galois theory and commutative ring extensions (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for galois theory and commutative ring extensions
- 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
13B10 Morphisms of commutative rings
Overview
13B10 develops morphisms of commutative rings within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Morphisms of commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for morphisms of 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
13B21 Integral dependence; going up, going down
Overview
13B21 develops integral dependence; going up, going down within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Integral dependence; going up, going down (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for integral dependence; going up, going down
- 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
13B22 Integral closure of commutative rings
Overview
13B22 develops integral closure of commutative rings within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Integral closure of commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for integral closure of 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
13B24 Going up; going down; lying over; flatness
Overview
13B24 develops going up; going down; lying over; flatness within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Going up; going down; lying over; flatness (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for going up; going down; lying over; flatness
- 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
13B25 Polynomials over commutative rings
Overview
13B25 develops polynomials over commutative rings within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Polynomials over commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for polynomials over 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
13B30 Rings of fractions and localization
Overview
13B30 develops rings of fractions and localization within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Rings of fractions and localization (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for rings of fractions and localization
- 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
13B35 Completion of commutative rings
Overview
13B35 develops completion of commutative rings within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Completion of commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for completion of 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
13B40 Étale and flat extensions; Henselization
Overview
13B40 develops etale and flat extensions; henselization within ring extensions and related structures. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.
Related Wikipedia Page
Etale and flat extensions; Henselization (Wikipedia)
Useful Links
Key Ideas
- Canonical examples and equivalent formulations for etale and flat extensions; henselization
- 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