English
Let R be a commutative ring of characteristic p, and let I be an ideal of R such that no nonzero natural number maps into I. Then the quotient ring R/I also has characteristic p.
Русский
Пусть R — коммутативное кольцо с характеристикой p, и пусть I будет идеалом R such that никакое ненулевое натуральное число не отображается в I. Тогда факторкольцо R/I тоже имеет характеристику p.
LaTeX
$$$ \\mathrm{CharP}(R/I) \\\\; p$$$
Lean4
/-- If an ideal does not contain any coercions of natural numbers other than zero, then its quotient
inherits the characteristic of the underlying ring. -/
theorem quotient' {R : Type*} [CommRing R] (p : ℕ) [CharP R p] (I : Ideal R) (h : ∀ x : ℕ, (x : R) ∈ I → (x : R) = 0) :
CharP (R ⧸ I) p :=
⟨fun x => by
rw [← cast_eq_zero_iff R p x, ← map_natCast (Ideal.Quotient.mk I)]
refine Ideal.Quotient.eq.trans (?_ : ↑x - 0 ∈ I ↔ _)
rw [sub_zero]
exact ⟨h x, fun h' => h'.symm ▸ I.zero_mem⟩⟩
