of_finite_primes — Mathlib

English

If a Dedekind domain has finitely many prime ideals, then it is a principal ideal domain.

Русский

Если у Дедекинд домена finitely много простых идеалов, то он является главной идеал-доменом.

LaTeX

$$$$\\text{IsPrincipalIdealRing}(R)$$$$

Lean4

/-- A Dedekind domain is a PID if its set of primes is finite. -/
theorem of_finite_primes [IsDedekindDomain R] (h : {I : Ideal R | I.IsPrime}.Finite) : IsPrincipalIdealRing R :=
  IsPrincipalIdealRing.of_finite_maximals <| h.subset fun _ hi ↦ hi.isPrime

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