Documentation

Mathlib.RingTheory.Localization.Integer

Integer elements of a localization #

Main definitions #

Implementation notes #

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

Tags #

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

def IsLocalization.IsInteger (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] (a : S) :

Given a : S, S a localization of R, IsInteger R a iff a is in the image of the localization map from R to S.

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    theorem IsLocalization.isInteger_add {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {a : S} {b : S} (ha : IsLocalization.IsInteger R a) (hb : IsLocalization.IsInteger R b) :
    theorem IsLocalization.isInteger_mul {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {a : S} {b : S} (ha : IsLocalization.IsInteger R a) (hb : IsLocalization.IsInteger R b) :
    theorem IsLocalization.isInteger_smul {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {a : R} {b : S} (hb : IsLocalization.IsInteger R b) :
    theorem IsLocalization.exists_integer_multiple' {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (a : S) :
    b, IsLocalization.IsInteger R (a * ↑(algebraMap R S) b)

    Each element a : S has an M-multiple which is an integer.

    This version multiplies a on the right, matching the argument order in LocalizationMap.surj.

    theorem IsLocalization.exists_integer_multiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (a : S) :
    b, IsLocalization.IsInteger R (b a)

    Each element a : S has an M-multiple which is an integer.

    This version multiplies a on the left, matching the argument order in the SMul instance.

    theorem IsLocalization.exist_integer_multiples {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ιS) :
    b, ∀ (i : ι), i sIsLocalization.IsInteger R (b f i)

    We can clear the denominators of a Finset-indexed family of fractions.

    theorem IsLocalization.exist_integer_multiples_of_finite {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} [Finite ι] (f : ιS) :
    b, ∀ (i : ι), IsLocalization.IsInteger R (b f i)

    We can clear the denominators of a finite indexed family of fractions.

    theorem IsLocalization.exist_integer_multiples_of_finset {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (s : Finset S) :
    b, ∀ (a : S), a sIsLocalization.IsInteger R (b a)

    We can clear the denominators of a finite set of fractions.

    noncomputable def IsLocalization.commonDenom {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ιS) :
    { x // x M }

    A choice of a common multiple of the denominators of a Finset-indexed family of fractions.

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      noncomputable def IsLocalization.integerMultiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ιS) (i : { x // x s }) :
      R

      The numerator of a fraction after clearing the denominators of a Finset-indexed family of fractions.

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      • One or more equations did not get rendered due to their size.
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        @[simp]
        theorem IsLocalization.map_integerMultiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ιS) (i : { x // x s }) :
        noncomputable def IsLocalization.commonDenomOfFinset {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (s : Finset S) :
        { x // x M }

        A choice of a common multiple of the denominators of a finite set of fractions.

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

          The finset of numerators after clearing the denominators of a finite set of fractions.

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