Theorems · Theorem · commutative algebra

Localization.mem_range_mapToFractionRing_iff

∀ {A : Type u_1} (K : Type u_2) [inst : CommRing A] (S : Submonoid A) [inst_1 : CommRing K] [inst_2 : Algebra A K]
  [inst_3 : IsFractionRing A K] (B : Type u_3) [inst_4 : CommRing B] [inst_5 : Algebra A B]
  [inst_6 : IsLocalization S B] (hS : S ≤ nonZeroDivisors A) (x : K),
  x ∈ (Localization.mapToFractionRing K S B hS).range ↔ ∃ a s, ∃ (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, ⋯⟩

Defined in

Mathlib.RingTheory.Localization.AsSubring

Cited by

1 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsFractionRingCommRingAlgebraIsLocalization