(Left) Ore sets and rings

This file contains results on left Ore sets for rings and monoids with zero.

References

• https://ncatlab.org/nlab/show/Ore+set

@[instance_reducible]

def OreLocalization.oreSetOfIsCancelMulZero {R : Type u_1} [MonoidWithZero R] [IsCancelMulZero R] {S : Submonoid R} (oreNum : R → ↥S → R) (oreDenom : R → ↥S → ↥S) (ore_eq : ∀ (r : R) (s : ↥S), ↑(oreDenom r s) * r = oreNum r s * ↑s) : OreSet S

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

Equations

• OreLocalization.oreSetOfIsCancelMulZero oreNum oreDenom ore_eq = { ore_right_cancel := ⋯, oreNum := oreNum, oreDenom := oreDenom, ore_eq := ore_eq }

@[instance_reducible]

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), ↑(oreDenom r s) * r = oreNum r s * ↑s) : 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.

Equations

• OreLocalization.oreSetOfNoZeroDivisors oreNum oreDenom ore_eq = OreLocalization.oreSetOfIsCancelMulZero oreNum oreDenom ore_eq

theorem OreLocalization.nonempty_oreSet_iff {R : Type u_1} [Monoid R] {S : Submonoid R} : Nonempty (OreSet S) ↔ (∀ (r₁ r₂ : R) (s : ↥S), r₁ * ↑s = r₂ * ↑s → ∃ (s' : ↥S), ↑s' * r₁ = ↑s' * r₂) ∧ ∀ (r : R) (s : ↥S), ∃ (r' : R), ∃ (s' : ↥S), ↑s' * r = r' * ↑s

theorem OreLocalization.nonempty_oreSet_iff_of_noZeroDivisors {R : Type u_1} [Ring R] [NoZeroDivisors R] {S : Submonoid R} : Nonempty (OreSet S) ↔ ∀ (r : R) (s : ↥S), ∃ (r' : R), ∃ (s' : ↥S), ↑s' * r = r' * ↑s