resolventSet_of_subsingleton — Mathlib

English

If A is a subsingleton ring, then the resolvent set of any a ∈ A is the entire base, i.e., resolventSet(R,a) = S univ (equivalently every r ∈ R lies in the resolvent set of a).

Русский

Если A —Подмножество (субоднородное) кольцо, то резольвентное множество любого элемента a ∈ A является всем основанием: resolventSet(R,a) = S univ.

LaTeX

$$$\operatorname{resolventSet}(R,a) = \mathrm{Set.univ}$$$

Lean4

@[simp]
theorem resolventSet_of_subsingleton [Subsingleton A] (a : A) : resolventSet R a = Set.univ := by
  simp_rw [resolventSet, Subsingleton.elim (algebraMap R A _ - a) 1, isUnit_one, Set.setOf_true]

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