Submonoid of inverses #

Main definitions #

IsLocalization.invSubmonoid M S is the submonoid of S = M⁻¹R consisting of inverses of each element x ∈ M

Implementation notes #

See Mathlib/RingTheory/Localization/Basic.lean for a design overview.

Tags #

localization, ring localization, commutative ring localization, characteristic predicate, commutative ring, field of fractions

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def IsLocalization.invSubmonoid {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] :

Submonoid S

The submonoid of S = M⁻¹R consisting of { 1 / x | x ∈ M }.

Instances For

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theorem IsLocalization.submonoid_map_le_is_unit {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] :

Submonoid.map (algebraMap R S) M ≤ IsUnit.submonoid S

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@[inline, reducible]

noncomputable abbrev IsLocalization.equivInvSubmonoid {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] :

{ x // x ∈ Submonoid.map (algebraMap R S) M } ≃* { x // x ∈ IsLocalization.invSubmonoid M S }

There is an equivalence of monoids between the image of M and invSubmonoid.

Instances For

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noncomputable def IsLocalization.toInvSubmonoid {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] :

{ x // x ∈ M } →* { x // x ∈ IsLocalization.invSubmonoid M S }

There is a canonical map from M to invSubmonoid sending x to 1 / x.

Instances For

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theorem IsLocalization.toInvSubmonoid_surjective {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] :

Function.Surjective ↑(IsLocalization.toInvSubmonoid M S)

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@[simp]

theorem IsLocalization.toInvSubmonoid_mul {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] (m : { x // x ∈ M }) :

↑(↑(IsLocalization.toInvSubmonoid M S) m) * ↑(algebraMap R S) ↑m = 1

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@[simp]

theorem IsLocalization.mul_toInvSubmonoid {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] (m : { x // x ∈ M }) :

↑(algebraMap R S) ↑m * ↑(↑(IsLocalization.toInvSubmonoid M S) m) = 1

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@[simp]

theorem IsLocalization.smul_toInvSubmonoid {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] (m : { x // x ∈ M }) :

m • ↑(↑(IsLocalization.toInvSubmonoid M S) m) = 1

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theorem IsLocalization.surj'' {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (z : S) :

∃ r m, z = r • ↑(↑(IsLocalization.toInvSubmonoid M S) m)

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theorem IsLocalization.toInvSubmonoid_eq_mk' {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (x : { x // x ∈ M }) :

↑(↑(IsLocalization.toInvSubmonoid M S) x) = IsLocalization.mk' S 1 x

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theorem IsLocalization.mem_invSubmonoid_iff_exists_mk' {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (x : S) :

x ∈ IsLocalization.invSubmonoid M S ↔ ∃ m, IsLocalization.mk' S 1 m = x

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theorem IsLocalization.span_invSubmonoid {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] :

Submodule.span R ↑(IsLocalization.invSubmonoid M S) = ⊤

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theorem IsLocalization.finiteType_of_monoid_fg {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] [Monoid.FG { x // x ∈ M }] :

Algebra.FiniteType R S