pNilradical_eq_bot_of_frobenius_inj

English

If Frobenius map Frobenius_{R,p} is injective for a ring R, then pNilradical R p = ⊥.

Русский

Если отображение Фробениуса Frobenius_{R,p} insulin инъективно, то pNilradical R p = ⊥.

LaTeX

$$$$ \\text{If } \\operatorname{Injective}(\\mathrm{frobenius}_{R,p}) \\text{ then } pNilradical(R,p)=\\bot. $$$$

Lean4

theorem pNilradical_eq_bot_of_frobenius_inj (R : Type*) [CommSemiring R] (p : ℕ) [ExpChar R p]
    (h : Function.Injective (frobenius R p)) : pNilradical R p = ⊥ :=
  bot_unique fun x ↦ by
    rw [mem_pNilradical, Ideal.mem_bot]
    exact fun ⟨n, _⟩ ↦ h.iterate n (by rwa [← coe_iterateFrobenius, map_zero])

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