Documentation

Mathlib.RingTheory.OreLocalization.OreSet

(Right) Ore sets #

This defines right Ore sets on arbitrary monoids.

References #

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class OreLocalization.OreSet {R : Type u_1} [inst : Monoid R] (S : Submonoid R) :
Type u_1

  • Common factors on the left can be turned into common factors on the right, a weak form of cancellability.

    ore_left_cancel : ∀ (r₁ r₂ : R) (s : { x // x ∈ S }), ↑s * r₁ = ↑s * r₂ → ∃ s', r₁ * ↑s' = r₂ * ↑s'

  • The Ore numerator of a fraction.

    oreNum : R → { x // x ∈ S } → R

  • The Ore denominator of a fraction.

    oreDenom : R → { x // x ∈ S } → { x // x ∈ S }

  • The Ore condition of a fraction, expressed in terms of oreNum and oreDenom.

    ore_eq : ∀ (r : R) (s : { x // x ∈ S }), r * ↑(oreDenom r s) = ↑s * oreNum r s

A submonoid S of a monoid R is (right) Ore if common factors on the left can be turned into common factors on the right, and if each pair of r : R and s : S admits an Ore numerator v : R and an Ore denominator u : S such that r * u = s * v.

Instances
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    theorem OreLocalization.ore_left_cancel {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst : OreLocalization.OreSet S] (r₁ : R) (r₂ : R) (s : { x // x ∈ S }) (h : ↑s * r₁ = ↑s * r₂) :
    ∃ s', r₁ * ↑s' = r₂ * ↑s'

    Common factors on the left can be turned into common factors on the right, a weak form of cancellability.

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    def OreLocalization.oreNum {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst : OreLocalization.OreSet S] (r : R) (s : { x // x ∈ S }) :
    R

    The Ore numerator of a fraction.

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    def OreLocalization.oreDenom {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst : OreLocalization.OreSet S] (r : R) (s : { x // x ∈ S }) :
    { x // x ∈ S }

    The Ore denominator of a fraction.

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    theorem OreLocalization.ore_eq {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst : OreLocalization.OreSet S] (r : R) (s : { x // x ∈ S }) :
    r * ↑(OreLocalization.oreDenom r s) = ↑s * OreLocalization.oreNum r s

    The Ore condition of a fraction, expressed in terms of oreNum and oreDenom.

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    def OreLocalization.oreCondition {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst : OreLocalization.OreSet S] (r : R) (s : { x // x ∈ S }) :
    (r' : R) ×' (s' : { x // x ∈ S }) ×' r * ↑s' = ↑s * r'

    The Ore condition bundled in a sigma type. This is useful in situations where we want to obtain both witnesses and the condition for a given fraction.

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    • One or more equations did not get rendered due to their size.
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    instance OreLocalization.oreSetBot {R : Type u_1} [inst : Monoid R] :
    OreLocalization.OreSet ⊥

    The trivial submonoid is an Ore set.

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    instance OreLocalization.oreSetComm {R : Type u_1} [inst : CommMonoid R] (S : Submonoid R) :
    OreLocalization.OreSet S

    Every submonoid of a commutative monoid is an Ore set.

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    def OreLocalization.oreSetOfCancelMonoidWithZero {R : Type u_1} [inst : CancelMonoidWithZero R] {S : Submonoid R} (oreNum : R → { x // x ∈ S } → R) (oreDenom : R → { x // x ∈ S } → { x // x ∈ S }) (ore_eq : ∀ (r : R) (s : { x // x ∈ S }), r * ↑(oreDenom r s) = ↑s * oreNum r s) :
    OreLocalization.OreSet S

    Cancellability in monoids with zeros can act as a replacement for the ore_left_cancel condition of an ore set.

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    • One or more equations did not get rendered due to their size.
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    def OreLocalization.oreSetOfNoZeroDivisors {R : Type u_1} [inst : Ring R] [inst : NoZeroDivisors R] {S : Submonoid R} (oreNum : R → { x // x ∈ S } → R) (oreDenom : R → { x // x ∈ S } → { x // x ∈ S }) (ore_eq : ∀ (r : R) (s : { x // x ∈ S }), r * ↑(oreDenom r s) = ↑s * oreNum r s) :
    OreLocalization.OreSet S

    In rings without zero divisors, the first (cancellability) condition is always fulfilled, it suffices to give a proof for the Ore condition itself.

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