English
In a DVR, an irreducible element is a uniformizer generating the maximal ideal.
Русский
В DVR нерелевантный элемент является единым генератором максимального идеала.
LaTeX
$$$\forall (R : Type) [\mathrm{CommRing R}] [IsDomain R] [IsDiscreteValuationRing R] (\varpi : R),
\mathrm{Irreducible } \varpi \iff \mathrm{maximalIdeal } R = \mathrm{Ideal}.span\{\varpi\}$$$
Lean4
theorem irreducible_of_span_eq_maximalIdeal {R : Type*} [CommSemiring R] [IsLocalRing R] [IsDomain R] (ϖ : R)
(hϖ : ϖ ≠ 0) (h : maximalIdeal R = Ideal.span { ϖ }) : Irreducible ϖ :=
by
have h2 : ¬IsUnit ϖ := show ϖ ∈ maximalIdeal R from h.symm ▸ Submodule.mem_span_singleton_self ϖ
refine ⟨h2, ?_⟩
intro a b hab
by_contra! h
obtain ⟨ha : a ∈ maximalIdeal R, hb : b ∈ maximalIdeal R⟩ := h
rw [h, mem_span_singleton'] at ha hb
rcases ha with ⟨a, rfl⟩
rcases hb with ⟨b, rfl⟩
rw [show a * ϖ * (b * ϖ) = ϖ * (ϖ * (a * b)) by ring] at hab
apply hϖ
apply eq_zero_of_mul_eq_self_right _ hab.symm
exact fun hh => h2 (isUnit_of_dvd_one ⟨_, hh.symm⟩)
