English
Let L be a finite separable extension of K = Frac(A), with A integrally closed and Noetherian. Then the integral closure C of A in L is Noetherian as an A-module, hence is a Noetherian ring in this setting.
Русский
Пусть L — конечное сепарабельноe расширение K = Frac(A), A интегрально замкнуто и Нетерированно. Тогда интегральное замыкание C A в L является Noetherian как A-модуль, следовательно, в данном контексте — Noetherian кольцо.
LaTeX
$$$$\\text{Module.Noetherian}_A(C)$$$$
Lean4
/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
Noetherian. -/
theorem isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRing A] : IsNoetherianRing C :=
isNoetherianRing_iff.mpr <| isNoetherian_of_tower A (IsIntegralClosure.isNoetherian A K L C)
