| import Mathlib | |
| namespace OAI | |
| namespace PiExponent.IdealFiniteGenerators | |
| theorem exists_generators {R : Type*} [CommRing R] [IsNoetherianRing R] | |
| (I : Ideal R) : ∃ m : ℕ, ∃ a : Fin m → I, | |
| Ideal.span (Set.range fun j => (a j).val) = I := by | |
| obtain ⟨m,b,hb⟩ := Submodule.fg_iff_exists_fin_generating_family.mp | |
| (IsNoetherian.noetherian I) | |
| let a : Fin m → I := fun j => ⟨b j, by | |
| rw [← hb] | |
| exact Submodule.subset_span ⟨j,rfl⟩⟩ | |
| exact ⟨m,a,hb⟩ | |
| end PiExponent.IdealFiniteGenerators | |
| end OAI | |