13Fxx Arithmetic and multiplicative structures
This subtopic studies arithmetic and multiplicative structures, including unique factorization, divisibility, and arithmetic properties of commutative rings.
Specific topics
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
Overview
13F05 develops dedekind, prufer, krull and mori rings and their generalizations in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Dedekind, Prufer, Krull and Mori rings and their generalizations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for dedekind, prufer, krull and mori rings and their generalizations
- 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
13F07 Euclidean rings and generalizations
Overview
13F07 develops euclidean rings and generalizations in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Euclidean rings and generalizations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for euclidean rings and generalizations
- 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
13F10 Principal ideal rings
Overview
13F10 develops principal ideal rings in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Principal ideal rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for principal ideal rings
- 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
13F15 Rings defined by factorization properties
Overview
13F15 develops rings defined by factorization properties in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Rings defined by factorization properties (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for rings defined by factorization properties
- 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
13F20 Polynomial rings and ideals
Overview
13F20 develops polynomial rings and ideals in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Polynomial rings and ideals (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for polynomial rings and ideals
- 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
13F25 Formal power series rings
Overview
13F25 develops formal power series rings in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Formal power series rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for formal power series rings
- 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
13F30 Valuation rings
Overview
13F30 develops valuation rings in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Valuation rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for valuation rings
- 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
13F35 Witt vectors and related rings
Overview
13F35 develops witt vectors and related rings in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Witt vectors and related rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for witt vectors and related rings
- 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
13F40 Excellent rings
Overview
13F40 develops excellent rings in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Excellent rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for excellent rings
- 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
13F45 Seminormal rings
Overview
13F45 develops seminormal rings in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Seminormal rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for seminormal rings
- 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
13F50 Rings with straightening laws, Hodge algebras
Overview
13F50 develops rings with straightening laws, hodge algebras in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Rings with straightening laws, Hodge algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for rings with straightening laws, hodge algebras
- 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
13F55 Stanley-Reisner face rings; simplicial complexes
Overview
13F55 develops stanley-reisner face rings; simplicial complexes in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Stanley-Reisner face rings; simplicial complexes (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for stanley-reisner face rings; simplicial complexes
- 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
13F60 Cluster algebras
Overview
13F60 develops cluster algebras in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Cluster algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for cluster algebras
- 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
13F65 Rings arising from non-commutative algebraic geometry
Overview
13F65 develops rings arising from non-commutative algebraic geometry in arithmetic and multiplicative structures of commutative rings. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Rings arising from non-commutative algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for rings arising from non-commutative algebraic geometry
- 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