disjoint_comap_iff — Mathlib

English

For any ideal J in S, disjointness with the comap is equivalent to J not being the top ideal.

Русский

Для любого идеала J в S дискриминантность относительно comap равна не равен ли J верхнему идеалу.

LaTeX

$$$\\operatorname{Disjoint}(M:\\mathrm{Set}R)(\\operatorname{Ideal.comap}(\\mathrm{algebraMap}\\;R\\;S)J) \\iff J \\neq \\top$$$

Lean4

theorem disjoint_comap_iff (J : Ideal S) : Disjoint (M : Set R) (J.comap (algebraMap R S)) ↔ J ≠ ⊤ :=
  by
  rw [← iff_not_comm, Set.not_disjoint_iff]
  constructor
  · rintro rfl
    exact ⟨1, M.one_mem, ⟨⟩⟩
  · rintro ⟨x, hxM, hxJ⟩
    exact J.eq_top_of_isUnit_mem hxJ (IsLocalization.map_units S ⟨x, hxM⟩)

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