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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | 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
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