nonempty — Mathlib

English

If R is Noetherian and M is nontrivial, then Ass_R(M) is nonempty.

Русский

Если R линеен и R — кольцо Безгранично, и M — не тривиален, тогда Ass_R(M) непусто.

LaTeX

$$$\\operatorname{Ass}_R(M) \\neq \\varnothing$ при $\\text{Noetherian}(R)$ и $\\text{Nontrivial}(M)$.$$

Lean4

theorem nonempty [IsNoetherianRing R] [Nontrivial M] : (associatedPrimes R M).Nonempty :=
  by
  obtain ⟨x, hx⟩ := exists_ne (0 : M)
  obtain ⟨P, hP, _⟩ := exists_le_isAssociatedPrime_of_isNoetherianRing R x hx
  exact ⟨P, hP⟩

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