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

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 }.

Equations

• IsLocalization.invSubmonoid M S = (Submonoid.map (algebraMap R S) M).leftInv

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

@[reducible, inline]

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] : ↥(Submonoid.map (algebraMap R S) M) ≃* ↥(invSubmonoid M S)

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

Equations

• IsLocalization.equivInvSubmonoid M S = ((Submonoid.map (algebraMap R S) M).leftInvEquiv ⋯).symm

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] : ↥M →* ↥(invSubmonoid M S)

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

Equations

• IsLocalization.toInvSubmonoid M S = (IsLocalization.equivInvSubmonoid M S).toMonoidHom.comp ((↑(algebraMap R S)).submonoidMap M)

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 ⇑(toInvSubmonoid M S)

@[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 : ↥M) : ↑((toInvSubmonoid M S) m) * (algebraMap R S) ↑m = 1

@[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 : ↥M) : (algebraMap R S) ↑m * ↑((toInvSubmonoid M S) m) = 1

@[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 : ↥M) : m • ↑((toInvSubmonoid M S) m) = 1

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 : R) (m : ↥M), z = r • ↑((toInvSubmonoid M S) m)

@[deprecated IsLocalization.surj' (since := "2026-08-20")]

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 : R) (m : ↥M), z = r • ↑((toInvSubmonoid M S) m)

Alias of IsLocalization.surj'.

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 : ↥M) : ↑((toInvSubmonoid M S) x) = mk' S 1 x

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 ∈ invSubmonoid M S ↔ ∃ (m : ↥M), mk' S 1 m = x

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 ↑(invSubmonoid M S) = ⊤

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 ↥M] : Algebra.FiniteType R S

instance IsLocalization.instFiniteTypeLocalizationOfFGSubtypeMemSubmonoid {R : Type u_3} {S : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Algebra.FiniteType R S] (M : Submonoid S) [Monoid.FG ↥M] : Algebra.FiniteType R (Localization M)