isNoetherian_coeIdeal — Mathlib

English

If R1 is Noetherian, then the Noetherian property holds for the coeIdeal of an ideal I.

Русский

Если R1 — Ноттерея, то coeIdeal I также Неотерминируем.

LaTeX

$$IsNoetherian R₁ (Ideal R₁) := IsNoetherian R₁ (coeIdeal I)$$

Lean4

theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) : IsNoetherian R₁ (I : FractionalIdeal R₁⁰ K) :=
  by
  rw [isNoetherian_iff]
  intro J hJ
  obtain ⟨J, rfl⟩ := le_one_iff_exists_coeIdeal.mp (le_trans hJ coeIdeal_le_one)
  exact (IsNoetherian.noetherian J).map _

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