Congruence relations on rings

This file contains basic results concerning congruence relations on rings, which extend Con and AddCon on monoids and additive monoids.

Most of the time you likely want to use the Ideal.Quotient API that is built on top of this.

Main Definitions

• RingCon R: the type of congruence relations respecting + and *.
• RingConGen r: the inductively defined smallest ring congruence relation containing a given binary relation.

TODO

• Use this for RingQuot too.
• Copy across more API from Con and AddCon in GroupTheory/Congruence.lean.

Scalar multiplication

The operation of scalar multiplication • descends naturally to the quotient.

instance RingCon.instSMulQuotient {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] (c : RingCon R) : SMul α c.Quotient

Equations

• c.instSMulQuotient = inferInstanceAs (SMul α c.Quotient)

@[simp]

theorem RingCon.coe_smul {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] (c : RingCon R) (a : α) (x : R) : ↑(a • x) = a • ↑x

instance RingCon.isScalarTower_right {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] (c : RingCon R) : IsScalarTower α c.Quotient c.Quotient

instance RingCon.smulCommClass {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] (c : RingCon R) : SMulCommClass α c.Quotient c.Quotient

instance RingCon.smulCommClass' {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] (c : RingCon R) : SMulCommClass c.Quotient α c.Quotient

instance RingCon.instDistribMulActionQuotientOfIsScalarTower {α : Type u_1} {R : Type u_2} [Monoid α] [NonAssocSemiring R] [DistribMulAction α R] [IsScalarTower α R R] (c : RingCon R) : DistribMulAction α c.Quotient

Equations

• c.instDistribMulActionQuotientOfIsScalarTower = DistribMulAction.mk ⋯ ⋯

instance RingCon.instMulSemiringActionQuotientOfIsScalarTower {α : Type u_1} {R : Type u_2} [Monoid α] [Semiring R] [MulSemiringAction α R] [IsScalarTower α R R] (c : RingCon R) : MulSemiringAction α c.Quotient

Equations

• c.instMulSemiringActionQuotientOfIsScalarTower = MulSemiringAction.mk ⋯ ⋯

Lattice structure

The API in this section is copied from Mathlib/GroupTheory/Congruence.lean

instance RingCon.instLE {R : Type u_2} [Add R] [Mul R] : LE (RingCon R)

For congruence relations c, d on a type M with multiplication and addition, c ≤ d iff ∀ x y ∈ M, x is related to y by d if x is related to y by c.

Equations

• RingCon.instLE = { le := fun (c d : RingCon R) => ∀ ⦃x y : R⦄, c x y → d x y }

theorem RingCon.le_def {R : Type u_2} [Add R] [Mul R] {c d : RingCon R} : c ≤ d ↔ ∀ {x y : R}, c x y → d x y

Definition of ≤ for congruence relations.

instance RingCon.instInfSet {R : Type u_2} [Add R] [Mul R] : InfSet (RingCon R)

The infimum of a set of congruence relations on a given type with multiplication and addition.

Equations

• RingCon.instInfSet = { sInf := fun (S : Set (RingCon R)) => { r := fun (x y : R) => ∀ c ∈ S, c x y, iseqv := ⋯, mul' := ⋯, add' := ⋯ } }

theorem RingCon.sInf_toSetoid {R : Type u_2} [Add R] [Mul R] (S : Set (RingCon R)) : (sInf S).toSetoid = sInf ((fun (x : RingCon R) => x.toSetoid) '' S)

The infimum of a set of congruence relations is the same as the infimum of the set's image under the map to the underlying equivalence relation.

@[simp]

theorem RingCon.coe_sInf {R : Type u_2} [Add R] [Mul R] (S : Set (RingCon R)) : ⇑(sInf S) = sInf (DFunLike.coe '' S)

The infimum of a set of congruence relations is the same as the infimum of the set's image under the map to the underlying binary relation.

@[simp]

theorem RingCon.coe_iInf {R : Type u_2} [Add R] [Mul R] {ι : Sort u_3} (f : ι → RingCon R) : ⇑(iInf f) = ⨅ (i : ι), ⇑(f i)

instance RingCon.instPartialOrder {R : Type u_2} [Add R] [Mul R] : PartialOrder (RingCon R)

Equations

• RingCon.instPartialOrder = PartialOrder.mk ⋯

instance RingCon.instCompleteLattice {R : Type u_2} [Add R] [Mul R] : CompleteLattice (RingCon R)

The complete lattice of congruence relations on a given type with multiplication and addition.

Equations

• RingCon.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[simp]

theorem RingCon.coe_top {R : Type u_2} [Add R] [Mul R] : ⇑⊤ = ⊤

@[simp]

theorem RingCon.coe_bot {R : Type u_2} [Add R] [Mul R] : ⇑⊥ = Eq

@[simp]

theorem RingCon.coe_inf {R : Type u_2} [Add R] [Mul R] {c d : RingCon R} : ⇑(c ⊓ d) = ⇑c ⊓ ⇑d

The infimum of two congruence relations equals the infimum of the underlying binary operations.

theorem RingCon.inf_iff_and {R : Type u_2} [Add R] [Mul R] {c d : RingCon R} {x y : R} : (c ⊓ d) x y ↔ c x y ∧ d x y

Definition of the infimum of two congruence relations.

instance RingCon.instNontrivial {R : Type u_2} [Add R] [Mul R] [Nontrivial R] : Nontrivial (RingCon R)

theorem RingCon.ringConGen_eq {R : Type u_2} [Add R] [Mul R] (r : R → R → Prop) : ringConGen r = sInf {s : RingCon R | ∀ (x y : R), r x y → s x y}

The inductively defined smallest congruence relation containing a binary relation r equals the infimum of the set of congruence relations containing r.

theorem RingCon.ringConGen_le {R : Type u_2} [Add R] [Mul R] {r : R → R → Prop} {c : RingCon R} (h : ∀ (x y : R), r x y → c x y) : ringConGen r ≤ c

The smallest congruence relation containing a binary relation r is contained in any congruence relation containing r.

theorem RingCon.ringConGen_mono {R : Type u_2} [Add R] [Mul R] {r s : R → R → Prop} (h : ∀ (x y : R), r x y → s x y) : ringConGen r ≤ ringConGen s

Given binary relations r, s with r contained in s, the smallest congruence relation containing s contains the smallest congruence relation containing r.

theorem RingCon.ringConGen_of_ringCon {R : Type u_2} [Add R] [Mul R] (c : RingCon R) : ringConGen ⇑c = c

Congruence relations equal the smallest congruence relation in which they are contained.

theorem RingCon.ringConGen_idem {R : Type u_2} [Add R] [Mul R] (r : R → R → Prop) : ringConGen ⇑(ringConGen r) = ringConGen r

The map sending a binary relation to the smallest congruence relation in which it is contained is idempotent.

theorem RingCon.sup_eq_ringConGen {R : Type u_2} [Add R] [Mul R] (c d : RingCon R) : c ⊔ d = ringConGen fun (x y : R) => c x y ∨ d x y

The supremum of congruence relations c, d equals the smallest congruence relation containing the binary relation 'x is related to y by c or d'.

theorem RingCon.sup_def {R : Type u_2} [Add R] [Mul R] {c d : RingCon R} : c ⊔ d = ringConGen (⇑c ⊔ ⇑d)

The supremum of two congruence relations equals the smallest congruence relation containing the supremum of the underlying binary operations.

theorem RingCon.sSup_eq_ringConGen {R : Type u_2} [Add R] [Mul R] (S : Set (RingCon R)) : sSup S = ringConGen fun (x y : R) => ∃ c ∈ S, c x y

The supremum of a set of congruence relations S equals the smallest congruence relation containing the binary relation 'there exists c ∈ S such that x is related to y by c'.

theorem RingCon.sSup_def {R : Type u_2} [Add R] [Mul R] {S : Set (RingCon R)} : sSup S = ringConGen (sSup (DFunLike.coe '' S))

The supremum of a set of congruence relations is the same as the smallest congruence relation containing the supremum of the set's image under the map to the underlying binary relation.

def RingCon.gi (R : Type u_2) [Add R] [Mul R] : GaloisInsertion ringConGen DFunLike.coe

There is a Galois insertion of congruence relations on a type with multiplication and addition R into binary relations on R.

Equations

• RingCon.gi R = { choice := fun (r : R → R → Prop) (_h : ⇑(ringConGen r) ≤ r) => ringConGen r, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }