Quotients of semirings

In this file, we directly define the quotient of a semiring by any relation, by building a bigger relation that represents the ideal generated by that relation.

We prove the universal properties of the quotient, and recommend avoiding relying on the actual definition, which is made irreducible for this purpose.

Since everything runs in parallel for quotients of R-algebras, we do that case at the same time.

inductive RingQuot.Rel {R : Type uR} [Semiring R] (r : R → R → Prop) : R → R → Prop

Given an arbitrary relation r on a ring, we strengthen it to a relation Rel r, such that the equivalence relation generated by Rel r has x ~ y if and only if x - y is in the ideal generated by elements a - b such that r a b.

• of {R : Type uR} [Semiring R] {r : R → R → Prop} ⦃x y : R⦄ (h : r x y) : Rel r x y
• add_left {R : Type uR} [Semiring R] {r : R → R → Prop} ⦃a b c : R⦄ : Rel r a b → Rel r (a + c) (b + c)
• mul_left {R : Type uR} [Semiring R] {r : R → R → Prop} ⦃a b c : R⦄ : Rel r a b → Rel r (a * c) (b * c)
• mul_right {R : Type uR} [Semiring R] {r : R → R → Prop} ⦃a b c : R⦄ : Rel r b c → Rel r (a * b) (a * c)

theorem RingQuot.Rel.add_right {R : Type uR} [Semiring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) : Rel r (a + b) (a + c)

theorem RingQuot.Rel.neg {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b : R⦄ (h : Rel r a b) : Rel r (-a) (-b)

theorem RingQuot.Rel.sub_left {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r a b) : Rel r (a - c) (b - c)

theorem RingQuot.Rel.sub_right {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) : Rel r (a - b) (a - c)

theorem RingQuot.Rel.smul {S : Type uS} [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {r : A → A → Prop} (k : S) ⦃a b : A⦄ (h : Rel r a b) : Rel r (k • a) (k • b)

def RingQuot.ringCon {R : Type uR} [Semiring R] (r : R → R → Prop) : RingCon R

EqvGen (RingQuot.Rel r) is a ring congruence.

Equations

• RingQuot.ringCon r = { r := Relation.EqvGen (RingQuot.Rel r), iseqv := ⋯, mul' := ⋯, add' := ⋯ }

theorem RingQuot.eqvGen_rel_eq {R : Type uR} [Semiring R] (r : R → R → Prop) : Relation.EqvGen (Rel r) = RingConGen.Rel r

structure RingQuot {R : Type uR} [Semiring R] (r : R → R → Prop) : Type uR

The quotient of a ring by an arbitrary relation.

• toQuot : Quot (Rel r)

def RingQuot.natCast {R : Type uR} [Semiring R] (r : R → R → Prop) (n : ℕ) : RingQuot r

The natCast function for RingQuot.

Equations

• RingQuot.natCast r n = { toQuot := Quot.mk (RingQuot.Rel r) ↑n }

@[implicit_reducible]

instance RingQuot.instNatCast {R : Type uR} [Semiring R] (r : R → R → Prop) : NatCast (RingQuot r)

Equations

• RingQuot.instNatCast r = { natCast := RingQuot.natCast r }

@[implicit_reducible]

instance RingQuot.instZero {R : Type uR} [Semiring R] (r : R → R → Prop) : Zero (RingQuot r)

Equations

• RingQuot.instZero r = { zero := { toQuot := Quot.mk (RingQuot.Rel r) 0 } }

@[implicit_reducible]

instance RingQuot.instOne {R : Type uR} [Semiring R] (r : R → R → Prop) : One (RingQuot r)

Equations

• RingQuot.instOne r = { one := { toQuot := Quot.mk (RingQuot.Rel r) 1 } }

@[implicit_reducible]

instance RingQuot.instAdd {R : Type uR} [Semiring R] (r : R → R → Prop) : Add (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance RingQuot.instMul {R : Type uR} [Semiring R] (r : R → R → Prop) : Mul (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance RingQuot.instNatPow {R : Type uR} [Semiring R] (r : R → R → Prop) : NatPow (RingQuot r)

Equations

• RingQuot.instNatPow r = { pow := fun (x : RingQuot r) (n : ℕ) => match x with | { toQuot := a } => { toQuot := Quot.lift (fun (a : R) => Quot.mk (RingQuot.Rel r) (a ^ n)) ⋯ a } }

@[implicit_reducible]

instance RingQuot.instNeg {R : Type uR} [Ring R] (r : R → R → Prop) : Neg (RingQuot r)

Equations

• RingQuot.instNeg r = { neg := fun (x : RingQuot r) => match x with | { toQuot := a } => { toQuot := Quot.map (fun (a : R) => -a) ⋯ a } }

@[implicit_reducible]

instance RingQuot.instSub {R : Type uR} [Ring R] (r : R → R → Prop) : Sub (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

def RingQuot.smul {R : Type uR} [Semiring R] {S : Type uS} [CommSemiring S] (r : R → R → Prop) [Algebra S R] (n : S) : RingQuot r → RingQuot r

The • function for RingQuot.

Equations

• RingQuot.smul r n { toQuot := a } = { toQuot := Quot.map (fun (a : R) => n • a) ⋯ a }

@[implicit_reducible]

instance RingQuot.instSMulOfAlgebra {R : Type uR} [Semiring R] {S : Type uS} [CommSemiring S] (r : R → R → Prop) [Algebra S R] : SMul S (RingQuot r)

Equations

• RingQuot.instSMulOfAlgebra r = { smul := RingQuot.smul r }

theorem RingQuot.zero_quot {R : Type uR} [Semiring R] (r : R → R → Prop) : { toQuot := Quot.mk (Rel r) 0 } = 0

theorem RingQuot.one_quot {R : Type uR} [Semiring R] (r : R → R → Prop) : { toQuot := Quot.mk (Rel r) 1 } = 1

theorem RingQuot.add_quot {R : Type uR} [Semiring R] (r : R → R → Prop) {a b : R} : { toQuot := Quot.mk (Rel r) a } + { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a + b) }

theorem RingQuot.mul_quot {R : Type uR} [Semiring R] (r : R → R → Prop) {a b : R} : { toQuot := Quot.mk (Rel r) a } * { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a * b) }

theorem RingQuot.pow_quot {R : Type uR} [Semiring R] (r : R → R → Prop) {a : R} {n : ℕ} : { toQuot := Quot.mk (Rel r) a } ^ n = { toQuot := Quot.mk (Rel r) (a ^ n) }

theorem RingQuot.neg_quot {R : Type uR} [Ring R] (r : R → R → Prop) {a : R} : -{ toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }

theorem RingQuot.sub_quot {R : Type uR} [Ring R] (r : R → R → Prop) {a b : R} : { toQuot := Quot.mk (Rel r) a } - { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a - b) }

theorem RingQuot.smul_quot {R : Type uR} [Semiring R] {S : Type uS} [CommSemiring S] (r : R → R → Prop) [Algebra S R] {n : S} {a : R} : n • { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (n • a) }

instance RingQuot.instIsScalarTower {R : Type uR} [Semiring R] {S : Type uS} [CommSemiring S] {T : Type uT} (r : R → R → Prop) [CommSemiring T] [SMul S T] [Algebra S R] [Algebra T R] [IsScalarTower S T R] : IsScalarTower S T (RingQuot r)

instance RingQuot.instSMulCommClass {R : Type uR} [Semiring R] {S : Type uS} [CommSemiring S] {T : Type uT} (r : R → R → Prop) [CommSemiring T] [Algebra S R] [Algebra T R] [SMulCommClass S T R] : SMulCommClass S T (RingQuot r)

@[implicit_reducible]

instance RingQuot.instAddCommMonoid {R : Type uR} [Semiring R] (r : R → R → Prop) : AddCommMonoid (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance RingQuot.instMonoidWithZero {R : Type uR} [Semiring R] (r : R → R → Prop) : MonoidWithZero (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance RingQuot.instSemiring {R : Type uR} [Semiring R] (r : R → R → Prop) : Semiring (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

def RingQuot.intCast {R : Type uR} [Ring R] (r : R → R → Prop) (z : ℤ) : RingQuot r

The intCast function for RingQuot.

Equations

• RingQuot.intCast r z = { toQuot := Quot.mk (RingQuot.Rel r) ↑z }

@[implicit_reducible]

instance RingQuot.instRing {R : Type uR} [Ring R] (r : R → R → Prop) : Ring (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance RingQuot.instCommSemiring {R : Type uR} [CommSemiring R] (r : R → R → Prop) : CommSemiring (RingQuot r)

Equations

• RingQuot.instCommSemiring r = { toSemiring := RingQuot.instSemiring r, mul_comm := ⋯ }

@[implicit_reducible]

instance RingQuot.instCommRing {R : Type uR} [CommRing R] (r : R → R → Prop) : CommRing (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance RingQuot.instInhabited {R : Type uR} [Semiring R] (r : R → R → Prop) : Inhabited (RingQuot r)

Equations

• RingQuot.instInhabited r = { default := 0 }

@[implicit_reducible]

instance RingQuot.instAlgebra {R : Type uR} [Semiring R] {S : Type uS} [CommSemiring S] [Algebra S R] (r : R → R → Prop) : Algebra S (RingQuot r)

Equations

• One or more equations did not get rendered due to their size.

theorem RingQuot.mkRingHom_def {R : Type u_1} [Semiring R] (r : R → R → Prop) : mkRingHom r = { toFun := fun (x : R) => { toQuot := Quot.mk (Rel r) x }, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[irreducible]

def RingQuot.mkRingHom {R : Type u_1} [Semiring R] (r : R → R → Prop) : R →+* RingQuot r

The quotient map from a ring to its quotient, as a homomorphism of rings.

Equations

• RingQuot.mkRingHom r = { toFun := fun (x : R) => { toQuot := Quot.mk (RingQuot.Rel r) x }, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

theorem RingQuot.mkRingHom_rel {R : Type uR} [Semiring R] {r : R → R → Prop} {x y : R} (w : r x y) : (mkRingHom r) x = (mkRingHom r) y

theorem RingQuot.mkRingHom_surjective {R : Type uR} [Semiring R] (r : R → R → Prop) : Function.Surjective ⇑(mkRingHom r)

theorem RingQuot.ringQuot_ext {R : Type uR} [Semiring R] {T : Type uT} [NonAssocSemiring T] {r : R → R → Prop} (f g : RingQuot r →+* T) (w : f.comp (mkRingHom r) = g.comp (mkRingHom r)) : f = g

theorem RingQuot.ringQuot_ext_iff {R : Type uR} [Semiring R] {T : Type uT} [NonAssocSemiring T] {r : R → R → Prop} {f g : RingQuot r →+* T} : f = g ↔ f.comp (mkRingHom r) = g.comp (mkRingHom r)

theorem RingQuot.preLift_def {R : Type u_1} [Semiring R] {T : Type u_2} [Semiring T] {r : R → R → Prop} {f : R →+* T} (h : ∀ ⦃x y : R⦄, r x y → f x = f y) : preLift h = { toFun := fun (x : RingQuot r) => Quot.lift ⇑f ⋯ x.toQuot, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[irreducible]

def RingQuot.preLift {R : Type u_1} [Semiring R] {T : Type u_2} [Semiring T] {r : R → R → Prop} {f : R →+* T} (h : ∀ ⦃x y : R⦄, r x y → f x = f y) : RingQuot r →+* T

Equations

• RingQuot.preLift h = { toFun := fun (x : RingQuot r) => Quot.lift ⇑f ⋯ x.toQuot, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

theorem RingQuot.lift_def {R : Type u_1} [Semiring R] {T : Type u_2} [Semiring T] {r : R → R → Prop} : lift = { toFun := fun (f : { f : R →+* T // ∀ ⦃x y : R⦄, r x y → f x = f y }) => preLift ⋯, invFun := fun (F : RingQuot r →+* T) => ⟨F.comp (mkRingHom r), ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[irreducible]

def RingQuot.lift {R : Type u_1} [Semiring R] {T : Type u_2} [Semiring T] {r : R → R → Prop} : { f : R →+* T // ∀ ⦃x y : R⦄, r x y → f x = f y } ≃ (RingQuot r →+* T)

Any ring homomorphism f : R →+* T which respects a relation r : R → R → Prop factors uniquely through a morphism RingQuot r →+* T.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem RingQuot.lift_mkRingHom_apply {R : Type uR} [Semiring R] {T : Type uT} [Semiring T] (f : R →+* T) {r : R → R → Prop} (w : ∀ ⦃x y : R⦄, r x y → f x = f y) (x : R) : (lift ⟨f, w⟩) ((mkRingHom r) x) = f x

theorem RingQuot.lift_unique {R : Type uR} [Semiring R] {T : Type uT} [Semiring T] (f : R →+* T) {r : R → R → Prop} (w : ∀ ⦃x y : R⦄, r x y → f x = f y) (g : RingQuot r →+* T) (h : g.comp (mkRingHom r) = f) : g = lift ⟨f, w⟩

theorem RingQuot.eq_lift_comp_mkRingHom {R : Type uR} [Semiring R] {T : Type uT} [Semiring T] {r : R → R → Prop} (f : RingQuot r →+* T) : f = lift ⟨f.comp (mkRingHom r), ⋯⟩

We now verify that in the case of a commutative ring, the RingQuot construction agrees with the quotient by the appropriate ideal.

def RingQuot.ringQuotToIdealQuotient {B : Type uR} [CommRing B] (r : B → B → Prop) : RingQuot r →+* B ⧸ Ideal.ofRel r

The universal ring homomorphism from RingQuot r to B ⧸ Ideal.ofRel r.

Equations

• RingQuot.ringQuotToIdealQuotient r = RingQuot.lift ⟨Ideal.Quotient.mk (Ideal.ofRel r), ⋯⟩

@[simp]

theorem RingQuot.ringQuotToIdealQuotient_apply {B : Type uR} [CommRing B] (r : B → B → Prop) (x : B) : (ringQuotToIdealQuotient r) ((mkRingHom r) x) = (Ideal.Quotient.mk (Ideal.ofRel r)) x

def RingQuot.idealQuotientToRingQuot {B : Type uR} [CommRing B] (r : B → B → Prop) : B ⧸ Ideal.ofRel r →+* RingQuot r

The universal ring homomorphism from B ⧸ Ideal.ofRel r to RingQuot r.

Equations

• RingQuot.idealQuotientToRingQuot r = Ideal.Quotient.lift (Ideal.ofRel r) (RingQuot.mkRingHom r) ⋯

@[simp]

theorem RingQuot.idealQuotientToRingQuot_apply {B : Type uR} [CommRing B] (r : B → B → Prop) (x : B) : (idealQuotientToRingQuot r) ((Ideal.Quotient.mk (Ideal.ofRel r)) x) = (mkRingHom r) x

def RingQuot.ringQuotEquivIdealQuotient {B : Type uR} [CommRing B] (r : B → B → Prop) : RingQuot r ≃+* B ⧸ Ideal.ofRel r

The ring equivalence between RingQuot r and (Ideal.ofRel r).quotient

Equations

• RingQuot.ringQuotEquivIdealQuotient r = RingEquiv.ofRingHom (RingQuot.ringQuotToIdealQuotient r) (RingQuot.idealQuotientToRingQuot r) ⋯ ⋯

@[irreducible]

def RingQuot.mkAlgHom (S : Type u_1) [CommSemiring S] {A : Type u_2} [Semiring A] [Algebra S A] (s : A → A → Prop) : A →ₐ[S] RingQuot s

The quotient map from an S-algebra to its quotient, as a homomorphism of S-algebras.

Equations

• RingQuot.mkAlgHom S s = let __src := RingQuot.mkRingHom s; { toRingHom := __src, commutes' := ⋯ }

theorem RingQuot.mkAlgHom_def (S : Type u_1) [CommSemiring S] {A : Type u_2} [Semiring A] [Algebra S A] (s : A → A → Prop) : mkAlgHom S s = let __src := mkRingHom s; { toRingHom := __src, commutes' := ⋯ }

@[simp]

theorem RingQuot.mkAlgHom_coe (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] (s : A → A → Prop) : ↑(mkAlgHom S s) = mkRingHom s

theorem RingQuot.mkAlgHom_rel (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {s : A → A → Prop} {x y : A} (w : s x y) : (mkAlgHom S s) x = (mkAlgHom S s) y

theorem RingQuot.mkAlgHom_surjective (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] (s : A → A → Prop) : Function.Surjective ⇑(mkAlgHom S s)

theorem RingQuot.ringQuot_ext' (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {B : Type u₄} [Semiring B] [Algebra S B] {s : A → A → Prop} (f g : RingQuot s →ₐ[S] B) (w : f.comp (mkAlgHom S s) = g.comp (mkAlgHom S s)) : f = g

theorem RingQuot.ringQuot_ext'_iff {S : Type uS} [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {B : Type u₄} [Semiring B] [Algebra S B] {s : A → A → Prop} {f g : RingQuot s →ₐ[S] B} : f = g ↔ f.comp (mkAlgHom S s) = g.comp (mkAlgHom S s)

@[irreducible]

def RingQuot.preLiftAlgHom (S : Type u_1) [CommSemiring S] {A : Type u_2} [Semiring A] [Algebra S A] {B : Type u_3} [Semiring B] [Algebra S B] {s : A → A → Prop} {f : A →ₐ[S] B} (h : ∀ ⦃x y : A⦄, s x y → f x = f y) : RingQuot s →ₐ[S] B

Equations

• RingQuot.preLiftAlgHom S h = { toFun := fun (x : RingQuot s) => Quot.lift ⇑f ⋯ x.toQuot, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

theorem RingQuot.preLiftAlgHom_def (S : Type u_1) [CommSemiring S] {A : Type u_2} [Semiring A] [Algebra S A] {B : Type u_3} [Semiring B] [Algebra S B] {s : A → A → Prop} {f : A →ₐ[S] B} (h : ∀ ⦃x y : A⦄, s x y → f x = f y) : preLiftAlgHom S h = { toFun := fun (x : RingQuot s) => Quot.lift ⇑f ⋯ x.toQuot, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

theorem RingQuot.liftAlgHom_def (S : Type u_1) [CommSemiring S] {A : Type u_2} [Semiring A] [Algebra S A] {B : Type u_3} [Semiring B] [Algebra S B] {s : A → A → Prop} : liftAlgHom S = { toFun := fun (f' : { f : A →ₐ[S] B // ∀ ⦃x y : A⦄, s x y → f x = f y }) => preLiftAlgHom S ⋯, invFun := fun (F : RingQuot s →ₐ[S] B) => ⟨F.comp (mkAlgHom S s), ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[irreducible]

def RingQuot.liftAlgHom (S : Type u_1) [CommSemiring S] {A : Type u_2} [Semiring A] [Algebra S A] {B : Type u_3} [Semiring B] [Algebra S B] {s : A → A → Prop} : { f : A →ₐ[S] B // ∀ ⦃x y : A⦄, s x y → f x = f y } ≃ (RingQuot s →ₐ[S] B)

Any S-algebra homomorphism f : A →ₐ[S] B which respects a relation s : A → A → Prop factors uniquely through a morphism RingQuot s →ₐ[S] B.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem RingQuot.liftAlgHom_mkAlgHom_apply (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {B : Type u₄} [Semiring B] [Algebra S B] (f : A →ₐ[S] B) {s : A → A → Prop} (w : ∀ ⦃x y : A⦄, s x y → f x = f y) (x : A) : ((liftAlgHom S) ⟨f, w⟩) ((mkAlgHom S s) x) = f x

theorem RingQuot.liftAlgHom_unique (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {B : Type u₄} [Semiring B] [Algebra S B] (f : A →ₐ[S] B) {s : A → A → Prop} (w : ∀ ⦃x y : A⦄, s x y → f x = f y) (g : RingQuot s →ₐ[S] B) (h : g.comp (mkAlgHom S s) = f) : g = (liftAlgHom S) ⟨f, w⟩

theorem RingQuot.eq_liftAlgHom_comp_mkAlgHom (S : Type uS) [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {B : Type u₄} [Semiring B] [Algebra S B] {s : A → A → Prop} (f : RingQuot s →ₐ[S] B) : f = (liftAlgHom S) ⟨f.comp (mkAlgHom S s), ⋯⟩

def RingQuot.algEquivQuotAlgEquiv {S : Type uS} [CommSemiring S] {A : Type uA} [Semiring A] [Algebra S A] {B : Type u₄} [Semiring B] [Algebra S B] (f : A ≃ₐ[S] B) (rel : A → A → Prop) : RingQuot rel ≃ₐ[S] RingQuot (Function.onFun rel ⇑f.symm)

If two S-algebras are S-equivalent and their quotients by a relation rel are defined, then their quotients are also S-equivalent.

(Special case of the third isomorphism theorem.)

Equations

• One or more equations did not get rendered due to their size.

def RingQuot.equivQuotEquiv {A : Type uA} [Semiring A] {B : Type u₄} [Semiring B] (f : A ≃+* B) (rel : A → A → Prop) : RingQuot rel ≃+* RingQuot (Function.onFun rel ⇑f.symm)

If two (semi)rings are equivalent and their quotients by a relation rel are defined, then their quotients are also equivalent.

(Special case of algEquivQuotAlgEquiv when S = ℕ, which in turn is a special case of the third isomorphism theorem.)

Equations

• RingQuot.equivQuotEquiv f rel = (RingQuot.algEquivQuotAlgEquiv (AlgEquiv.ofRingEquiv ⋯) rel).toRingEquiv