index_pow_le — Mathlib

English

If R is Noetherian, then for any ideal I, the quotient R/I is Noetherian.

Русский

Если R же Неттерское кольцо, то R/I тоже Неттерское для любого идеала I.

LaTeX

$$$$\text{IsNoetherianRing}(R/I).$$$$

Lean4

theorem index_pow_le (s : Finset R) (hs : Ideal.span s = I) [Finite (R ⧸ I)] (n) :
    (I ^ n).toAddSubgroup.index ≤ I.toAddSubgroup.index ^ ∑ i ∈ Finset.range n, s.card ^ i :=
  by
  have := Ideal.finite_quotient_pow ⟨s, hs⟩
  induction n with
  | zero => simp
  | succ n IH =>
    refine (Submodule.index_smul_le (I ^ n) s hs).trans ?_
    refine (Nat.mul_le_mul (Nat.pow_le_pow_left IH _) le_rfl).trans ?_
    rw [← pow_mul, ← pow_succ, geom_sum_succ, mul_comm]

Похожие утверждения