Theorems · Definition · commutative algebra

IsLocalization.AtPrime.orderIsoOfPrime

{R : Type u_1} →
  [inst : CommSemiring R] →
    (S : Type u_2) →
      [inst_1 : CommSemiring S] →
        [inst_2 : Algebra R S] →
          (I : Ideal R) →
            [hI : I.IsPrime] → [IsLocalization.AtPrime S I] → { p // p.IsPrime } ≃o { p // p.IsPrime ∧ p ≤ I }

The prime ideals in the localization of a commutative ring at a prime ideal I are in order-preserving bijection with the prime ideals contained in I.

Defined in

Mathlib.RingTheory.Localization.AtPrime.Basic

Cited by

4 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraIdeal.IsPrimeIsLocalization.AtPrime