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
Morphisms
Internally
Elements
- Unit, Zero-divisor
- Irreducible element, Prime element, Associate elements
- Algebraic element, Integral element
- GCD