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Mathlib.RingTheory.DedekindDomain.PID

Proving a Dedekind domain is a PID #

This file contains some results that we can use to show all ideals in a Dedekind domain are principal.

Main results #

theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {R : Type u_1} [CommRing R] {P : Ideal R} (hP : Ideal.IsPrime P) [IsDomain R] [IsDedekindDomain R] {x : R} (x_mem : x P) (hxP2 : xP ^ 2) (hxQ : ∀ (Q : Ideal R), Ideal.IsPrime QQ PxQ) :

Let P be a prime ideal, x ∈ P \ P² and x ∉ Q for all prime ideals Q ≠ P. Then P is generated by x.

theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S nonZeroDivisors R) (hf : Set.Finite {I : Ideal R | Ideal.IsMaximal I}) (I : FractionalIdeal S A) (I' : FractionalIdeal S A) (hinv : I * I' = 1) :

An invertible fractional ideal of a commutative ring with finitely many maximal ideals is principal.

https://math.stackexchange.com/a/95857

An invertible ideal in a commutative ring with finitely many maximal ideals is principal.

https://math.stackexchange.com/a/95857

A Dedekind domain is a PID if its set of primes is finite.

If p is a prime in the Dedekind domain R, S an extension of R and Sₚ the localization of S at p, then all primes in Sₚ are factors of the image of p in Sₚ.

Let p be a prime in the Dedekind domain R and S be an integral extension of R, then the localization Sₚ of S at p is a PID.