isUnit_or_isNilpotent_of_isMaximal_isNilpotent

English

In a local ring extension like TrivSqZeroExt, every element is either a unit or nilpotent under suitable hypotheses (locality).

Русский

В локальном колесе расширения TrivSqZeroExt каждый элемент либо единичен, либо нилпотентен при подходящих предположениях (локальность).

LaTeX

$$$IsUnit(a) \\lor IsNilpotent(a)$$$

Lean4

theorem isUnit_or_isNilpotent_of_isMaximal_isNilpotent [CommSemiring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M]
    [IsCentralScalar R M] (h : ∀ I : Ideal R, I.IsMaximal → IsNilpotent I) (a : TrivSqZeroExt R M) :
    IsUnit a ∨ IsNilpotent a :=
  by
  rw [isUnit_iff_isUnit_fst, isNilpotent_iff_isNilpotent_fst]
  refine (em _).imp_right fun ha ↦ ?_
  obtain ⟨I, hI, haI⟩ := exists_max_ideal_of_mem_nonunits (mem_nonunits_iff.mpr ha)
  refine (h _ hI).imp fun n hn ↦ ?_
  exact hn.le (Ideal.pow_mem_pow haI _)

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