ACC on ideals. Hilbert basis: R Noetherian ⇒ R[x] Noetherian. Primary ideals, Artin–Rees.
graph TD
D1["Def: Noetherian\nACC on ideals"]
D2["Def: Finitely generated\nideal I=(a₁,...,aₙ)"]
D3["Def: Hilbert basis\nR Noeth ⇒ R[x] Noeth"]
D4["Def: Artinian\nDCC on ideals"]
T1["Thm: I fg ⇔ ACC\nNoetherian equiv"]
T2["Thm: Hilbert basis\nR[x] Noeth"]
T3["Thm: Quotient Noeth\nR/I Noeth"]
T4["Thm: Localization\nS⁻¹R Noeth"]
T5["Thm: Artin–Rees\nIⁿ∩J stabilizes"]
D1 --> T1
D1 --> T2
D2 --> T1
D3 --> T2
T2 --> T3
T3 --> T4
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 14
- Edges: 11
Frontier: math.AC