File: Mathlib/RingTheory/Localization/AsSubring.lean

full_name: Localization.map_isUnit_of_le
code: theorem map_isUnit_of_le (hS : S ≤ A⁰) (s : S) : IsUnit (algebraMap A K s) 
goal: IsUnit ((algebraMap A K) ↑s)

full_name: Localization.mapToFractionRing
code: /-- The canonical map from a localization of `A` at `S` to the fraction ring
  of `A`, given that `S ≤ A⁰`. -/
noncomputable def mapToFractionRing (B : Type*) [CommRing B] [Algebra A B] [IsLocalization S B]
    (hS : S ≤ A⁰) : B →ₐ[A] K 
goal: B →ₐ[A] K

full_name: Localization.mapToFractionRing_apply
code: @[simp]
theorem mapToFractionRing_apply {B : Type*} [CommRing B] [Algebra A B] [IsLocalization S B]
    (hS : S ≤ A⁰) (b : B) :
    mapToFractionRing K S B hS b = IsLocalization.lift (map_isUnit_of_le K S hS) b 
goal: (Localization.mapToFractionRing K S B hS) b = (IsLocalization.lift ⋯) b

full_name: Localization.mem_range_mapToFractionRing_iff
code: theorem mem_range_mapToFractionRing_iff (B : Type*) [CommRing B] [Algebra A B] [IsLocalization S B]
    (hS : S ≤ A⁰) (x : K) :
    x ∈ (mapToFractionRing K S B hS).range ↔
      ∃ (a s : A) (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, hS hs⟩ 
goal: x ∈ (Localization.mapToFractionRing K S B hS).range ↔ ∃ a s, ∃ (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, ⋯⟩

full_name: Localization.isLocalization_range_mapToFractionRing
code: instance isLocalization_range_mapToFractionRing (B : Type*) [CommRing B] [Algebra A B]
    [IsLocalization S B] (hS : S ≤ A⁰) : IsLocalization S (mapToFractionRing K S B hS).range 
goal: IsLocalization S ↥(Localization.mapToFractionRing K S B hS).range

full_name: Localization.isFractionRing_range_mapToFractionRing
code: instance isFractionRing_range_mapToFractionRing (B : Type*) [CommRing B] [Algebra A B]
    [IsLocalization S B] (hS : S ≤ A⁰) : IsFractionRing (mapToFractionRing K S B hS).range K 
goal: IsFractionRing (↥(Localization.mapToFractionRing K S B hS).range) K

full_name: Localization.subalgebra
code: /-- Given a commutative ring `A` with fraction ring `K`, and a submonoid `S` of `A` which
contains no zero divisor, this is the localization of `A` at `S`, considered as
a subalgebra of `K` over `A`.

The carrier of this subalgebra is defined as the set of all `x : K` of the form
`IsLocalization.mk' K a ⟨s, _⟩`, where `s ∈ S`.
-/
noncomputable def subalgebra (hS : S ≤ A⁰) : Subalgebra A K 
goal: Subalgebra A K

full_name: Localization.subalgebra.isLocalization_subalgebra
code: instance isLocalization_subalgebra : IsLocalization S (subalgebra K S hS) 
goal: IsLocalization S ↥(Localization.subalgebra K S hS)

full_name: Localization.subalgebra.isFractionRing
code: instance isFractionRing : IsFractionRing (subalgebra K S hS) K 
goal: IsFractionRing (↥(Localization.subalgebra K S hS)) K

full_name: Localization.subalgebra.mem_range_mapToFractionRing_iff_ofField
code: theorem mem_range_mapToFractionRing_iff_ofField (B : Type*) [CommRing B] [Algebra A B]
    [IsLocalization S B] (x : K) :
    x ∈ (mapToFractionRing K S B hS).range ↔
      ∃ (a s : A) (_ : s ∈ S), x = algebraMap A K a * (algebraMap A K s)⁻¹ 
goal: x ∈ (Localization.mapToFractionRing K S B hS).range ↔
  ∃ a s, ∃ (_ : s ∈ S), x = (algebraMap A K) a * ((algebraMap A K) s)⁻¹

full_name: Localization.subalgebra.ofField
code: /-- Given a domain `A` with fraction field `K`, and a submonoid `S` of `A` which
contains no zero divisor, this is the localization of `A` at `S`, considered as
a subalgebra of `K` over `A`.

The carrier of this subalgebra is defined as the set of all `x : K` of the form
`algebraMap A K a * (algebraMap A K s)⁻¹` where `a s : A` and `s ∈ S`.
-/
noncomputable def ofField : Subalgebra A K 
goal: Subalgebra A K

full_name: Localization.subalgebra.isLocalization_ofField
code: instance isLocalization_ofField : IsLocalization S (subalgebra.ofField K S hS) 
goal: IsLocalization S ↥(Localization.subalgebra.ofField K S hS)

full_name: Localization.subalgebra.isFractionRing_ofField
code: instance isFractionRing_ofField : IsFractionRing (subalgebra.ofField K S hS) K 
goal: IsFractionRing (↥(Localization.subalgebra.ofField K S hS)) K