English
Under suitable Char-zero quotient conditions, a positive natural number n maps to a unit in the ring R.
Русский
При заданном условии нули характеристик для квадратов по идеалам, число n превращается в единицу в кольце R.
LaTeX
$$$\\forall n \\in \\mathbb{N}_{>0}, \\operatorname{IsUnit}(n: R)$$$
Lean4
/-- `ℚ`-algebra implies equal characteristic. -/
theorem of_algebraRat [Algebra ℚ R] : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) :=
by
intro I hI
constructor
intro a b h_ab
contrapose! hI
refine I.eq_top_of_isUnit_mem ?_ (IsUnit.map (algebraMap ℚ R) (IsUnit.mk0 (a - b : ℚ) ?_))
· simpa only [← Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]
simpa only [Ne, sub_eq_zero] using (@Nat.cast_injective ℚ _ _).ne hI
