exists_max_ideal_of_mem_nonunits — Mathlib

English

In a commutative semiring, every nonunit a lies in some maximal ideal.

Русский

В коммутативном полуг ring существует максимальный идеал, содержащий неединицу a.

LaTeX

$$∃ I : Ideal α, I.IsMaximal ∧ a ∈ I$$

Lean4

theorem exists_max_ideal_of_mem_nonunits [CommSemiring α] (h : a ∈ nonunits α) : ∃ I : Ideal α, I.IsMaximal ∧ a ∈ I :=
  by
  have : Ideal.span ({ a } : Set α) ≠ ⊤ := by
    intro H
    rw [Ideal.span_singleton_eq_top] at H
    contradiction
  rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩
  use I, Imax
  apply H
  apply Ideal.subset_span
  exact Set.mem_singleton a

Похожие утверждения