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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} [Monoid R] (S : Submonoid R) :
Type u_1

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.

  • ore_left_cancel : ∀ (r₁ r₂ : R) (s : ↥S), ↑s * r₁ = ↑s * r₂ → ∃ (s' : ↥S), r₁ * ↑s' = r₂ * ↑s'

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

  • oreNum : R → ↥S → R

    The Ore numerator of a fraction.

  • oreDenom : R → ↥S → ↥S

    The Ore denominator of a fraction.

  • ore_eq : ∀ (r : R) (s : ↥S), r * ↑(OreLocalization.OreSet.oreDenom r s) = ↑s * OreLocalization.OreSet.oreNum r s

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

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

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

    source
    def OreLocalization.oreNum {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] (r : R) (s : ↥S) :
    R

    The Ore numerator of a fraction.

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

      The Ore denominator of a fraction.

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

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

        source
        def OreLocalization.oreCondition {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] (r : R) (s : ↥S) :
        (r' : R) ×' (s' : ↥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.
        Instances For
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          instance OreLocalization.oreSetBot {R : Type u_1} [Monoid R] :
          OreLocalization.OreSet ⊥

          The trivial submonoid is an Ore set.

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          • One or more equations did not get rendered due to their size.
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          instance OreLocalization.oreSetComm {R : Type u_2} [CommMonoid R] (S : Submonoid R) :
          OreLocalization.OreSet S

          Every submonoid of a commutative monoid is 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.oreSetOfCancelMonoidWithZero {R : Type u_1} [CancelMonoidWithZero R] {S : Submonoid R} (oreNum : R → ↥S → R) (oreDenom : R → ↥S → ↥S) (ore_eq : ∀ (r : R) (s : ↥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} [Ring R] [NoZeroDivisors R] {S : Submonoid R} (oreNum : R → ↥S → R) (oreDenom : R → ↥S → ↥S) (ore_eq : ∀ (r : R) (s : ↥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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