Abstract algebra MOC

Ring theory MOC

Objects

The graph below is a non-exhaustive classification of rings

graph TD
    NumField[Number field]:::internal-link
    ACField[Algebraically closed field]:::internal-link
    Field:::internal-link
    NumField --> Field
    ACField --> Field

    Division[Division ring]:::internal-link
    Simple[Simple ring]:::internal-link
    PIR[Principal ideal ring]:::internal-link
    Noetherian[Noetherian ring]:::internal-link
    Field --> Division --> Simple --> PIR --> Noetherian --> Ring

    EDomain[Euclidean domain]:::internal-link
    PID[Principal ideal domain]:::internal-link
    Field --> EDomain --> PID --> PIR
    
    Dedekind[Dedekind domain]:::internal-link
    Dedekind --> Noetherian
    Dedekind --> IDomain
    PID --> Dedekind

    UFD[Unique factorization domain]:::internal-link
    GCDDomain[GCD domain]:::internal-link
    IDomain[Integral domain]:::internal-link
    Commutative[Commutative ring]:::internal-link
    PID --> UFD --> GCDDomain --> IDomain --> Commutative --> Ring

    Ring:::internal-link
    Rng:::internal-link
    Rig:::internal-link
    Ring --> Rng
    Ring --> Rig

Examples

• Modular arithmetic

Morphisms

• Rng homomorphism, Ring homomorphism

Internally

• Characteristic
• Krull dimension

Elements

• Unit, Zero-divisor
• Irreducible element, Prime element, Associate elements
• Algebraic element, Integral element
• GCD

Ideals

• Ideal
• Maximal ideal, Prime ideal

Externally

• Subrng, Subring, Field extension
• Ideal, Quotient ring

Related

• Module theory MOC
• Field theory MOC
• Algebraic number theory MOC

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