Ideal quotients

This file defines ideal quotients as a special case of submodule quotients and proves some basic results about these quotients.

See Algebra.RingQuot for quotients of non-commutative rings.

Main definitions

• Ideal.instHasQuotient: the quotient of a commutative ring R by an ideal I : Ideal R
• Ideal.Quotient.commRing: the ring structure of the ideal quotient
• Ideal.Quotient.mk: map an element of R to the quotient R ⧸ I
• Ideal.Quotient.lift: turn a map R → S into a map R ⧸ I → S
• Ideal.quotEquivOfEq: quotienting by equal ideals gives isomorphic rings

instance Ideal.instHasQuotient {R : Type u} [Ring R] : HasQuotient R (Ideal R)

The quotient R/I of a ring R by an ideal I, defined to equal the quotient of I as an R-submodule of R.

Equations

• Ideal.instHasQuotient = Submodule.hasQuotient

instance Ideal.instHasQuotient_1 {R : Type u_1} [CommRing R] : HasQuotient R (Ideal R)

Shortcut instance for commutative rings.

Equations

• Ideal.instHasQuotient_1 = inferInstance

instance Ideal.Quotient.one {R : Type u} [Ring R] (I : Ideal R) : One (R ⧸ I)

Equations

• Ideal.Quotient.one I = { one := Submodule.Quotient.mk 1 }

def Ideal.Quotient.ringCon {R : Type u} [Ring R] (I : Ideal R) [I.IsTwoSided] : RingCon R

On Ideals, Submodule.quotientRel is a ring congruence.

Equations

• Ideal.Quotient.ringCon I = { toSetoid := (QuotientAddGroup.con (Submodule.toAddSubgroup I)).toSetoid, mul' := ⋯, add' := ⋯ }

instance Ideal.Quotient.ring {R : Type u} [Ring R] (I : Ideal R) [I.IsTwoSided] : Ring (R ⧸ I)

Equations

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

instance Ideal.Quotient.commRing {R : Type u_1} [CommRing R] (I : Ideal R) : CommRing (R ⧸ I)

Equations

• Ideal.Quotient.commRing I = { toRing := Ideal.Quotient.ring I, mul_comm := ⋯ }

instance Ideal.Quotient.instRingQuotient {R : Type u_1} [CommRing R] (I : Ideal R) : Ring (R ⧸ I)

Equations

• Ideal.Quotient.instRingQuotient I = Ideal.Quotient.ring I

instance Ideal.Quotient.commSemiring {R : Type u_1} [CommRing R] (I : Ideal R) : CommSemiring (R ⧸ I)

Equations

• Ideal.Quotient.commSemiring I = CommRing.toCommSemiring

instance Ideal.Quotient.semiring {R : Type u_1} [CommRing R] (I : Ideal R) : Semiring (R ⧸ I)

Equations

• Ideal.Quotient.semiring I = (Ideal.Quotient.commSemiring I).toSemiring

def Ideal.Quotient.mk {R : Type u} [Ring R] (I : Ideal R) [I.IsTwoSided] : R →+* R ⧸ I

The ring homomorphism from a ring R to a quotient ring R/I.

Equations

• Ideal.Quotient.mk I = { toFun := fun (a : R) => Submodule.Quotient.mk a, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

instance Ideal.Quotient.instCoeQuotient {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] : Coe R (R ⧸ I)

Equations

• Ideal.Quotient.instCoeQuotient = { coe := ⇑(Ideal.Quotient.mk I) }

theorem Ideal.Quotient.ringHom_ext {R : Type u} [Ring R] {I : Ideal R} {S : Type v} [I.IsTwoSided] [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) : f = g

Two RingHoms from the quotient by an ideal are equal if their compositions with Ideal.Quotient.mk' are equal.

See note [partially-applied ext lemmas].

theorem Ideal.Quotient.ringHom_ext_iff {R : Type u} [Ring R] {I : Ideal R} {S : Type v} [I.IsTwoSided] [NonAssocSemiring S] {f g : R ⧸ I →+* S} : f = g ↔ f.comp (mk I) = g.comp (mk I)

instance Ideal.Quotient.instNonemptyQuotient {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] : Nonempty (R ⧸ I)

theorem Ideal.Quotient.eq {R : Type u} [Ring R] {I : Ideal R} {x y : R} [I.IsTwoSided] : (mk I) x = (mk I) y ↔ x - y ∈ I

@[simp]

theorem Ideal.Quotient.mk_eq_mk {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] (x : R) : Submodule.Quotient.mk x = (mk I) x

theorem Ideal.Quotient.eq_zero_iff_mem {R : Type u} [Ring R] {I : Ideal R} {a : R} [I.IsTwoSided] : (mk I) a = 0 ↔ a ∈ I

theorem Ideal.Quotient.mk_eq_mk_iff_sub_mem {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] (x y : R) : (mk I) x = (mk I) y ↔ x - y ∈ I

@[simp]

theorem Ideal.Quotient.mk_out {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] (x : R ⧸ I) : (mk I) (Quotient.out x) = x

theorem Ideal.Quotient.mk_surjective {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] : Function.Surjective ⇑(mk I)

instance Ideal.Quotient.instRingHomSurjectiveQuotientMk {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] : RingHomSurjective (mk I)

theorem Ideal.Quotient.quotient_ring_saturate {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] (s : Set R) : ⇑(mk I) ⁻¹' (⇑(mk I) '' s) = ⋃ (x : ↥I), (fun (y : R) => ↑x + y) '' s

If I is an ideal of a commutative ring R, if q : R → R/I is the quotient map, and if s ⊆ R is a subset, then q⁻¹(q(s)) = ⋃ᵢ(i + s), the union running over all i ∈ I.

def Ideal.Quotient.lift {R : Type u} [Ring R] (I : Ideal R) {S : Type v} [I.IsTwoSided] [Semiring S] (f : R →+* S) (H : ∀ a ∈ I, f a = 0) : R ⧸ I →+* S

Given a ring homomorphism f : R →+* S sending all elements of an ideal to zero, lift it to the quotient by this ideal.

Equations

• Ideal.Quotient.lift I f H = { toFun := (↑(QuotientAddGroup.lift (Submodule.toAddSubgroup I) f.toAddMonoidHom H)).toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Ideal.Quotient.lift_mk {R : Type u} [Ring R] (I : Ideal R) {a : R} {S : Type v} [I.IsTwoSided] [Semiring S] (f : R →+* S) (H : ∀ a ∈ I, f a = 0) : (lift I f H) ((mk I) a) = f a

theorem Ideal.Quotient.lift_comp_mk {R : Type u} [Ring R] (I : Ideal R) {S : Type v} [I.IsTwoSided] [Semiring S] (f : R →+* S) (H : ∀ a ∈ I, f a = 0) : (lift I f H).comp (mk I) = f

theorem Ideal.Quotient.lift_surjective_of_surjective {R : Type u} [Ring R] (I : Ideal R) {S : Type v} [I.IsTwoSided] [Semiring S] {f : R →+* S} (H : ∀ a ∈ I, f a = 0) (hf : Function.Surjective ⇑f) : Function.Surjective ⇑(lift I f H)

def Ideal.Quotient.factor {R : Type u} [Ring R] {S T : Ideal R} [S.IsTwoSided] [T.IsTwoSided] (H : S ≤ T) : R ⧸ S →+* R ⧸ T

The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.

This is the Ideal.Quotient version of Quot.Factor

When the two ideals are of the form I^m and I^n and n ≤ m, please refer to the dedicated version Ideal.Quotient.factorPow.

Equations

• Ideal.Quotient.factor H = Ideal.Quotient.lift S (Ideal.Quotient.mk T) ⋯

@[simp]

theorem Ideal.Quotient.factor_mk {R : Type u} [Ring R] {S T : Ideal R} [S.IsTwoSided] [T.IsTwoSided] (H : S ≤ T) (x : R) : (factor H) ((mk S) x) = (mk T) x

@[simp]

theorem Ideal.Quotient.factor_eq {R : Type u} [Ring R] {S : Ideal R} [S.IsTwoSided] : factor ⋯ = RingHom.id (R ⧸ S)

@[simp]

theorem Ideal.Quotient.factor_comp_mk {R : Type u} [Ring R] {S T : Ideal R} [S.IsTwoSided] [T.IsTwoSided] (H : S ≤ T) : (factor H).comp (mk S) = mk T

@[simp]

theorem Ideal.Quotient.factor_comp {R : Type u} [Ring R] {S T U : Ideal R} [S.IsTwoSided] [T.IsTwoSided] [U.IsTwoSided] (H1 : S ≤ T) (H2 : T ≤ U) : (factor H2).comp (factor H1) = factor ⋯

@[simp]

theorem Ideal.Quotient.factor_comp_apply {R : Type u} [Ring R] {S T U : Ideal R} [S.IsTwoSided] [T.IsTwoSided] [U.IsTwoSided] (H1 : S ≤ T) (H2 : T ≤ U) (x : R ⧸ S) : (factor H2) ((factor H1) x) = (factor ⋯) x

theorem Ideal.Quotient.factor_surjective {R : Type u} [Ring R] {S T : Ideal R} [S.IsTwoSided] [T.IsTwoSided] (H : S ≤ T) : Function.Surjective ⇑(factor H)

def Ideal.quotEquivOfEq {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) : R ⧸ I ≃+* R ⧸ J

Quotienting by equal ideals gives equivalent rings.

See also Submodule.quotEquivOfEq and Ideal.quotientEquivAlgOfEq.

Equations

• Ideal.quotEquivOfEq h = { toFun := (↑(Submodule.quotEquivOfEq I J h)).toFun, invFun := (Submodule.quotEquivOfEq I J h).invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯ }

@[simp]

theorem Ideal.quotEquivOfEq_mk {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) (x : R) : (quotEquivOfEq h) ((Quotient.mk I) x) = (Quotient.mk J) x

@[simp]

theorem Ideal.quotEquivOfEq_symm {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) : (quotEquivOfEq h).symm = quotEquivOfEq ⋯