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.Quotient: the quotient of a commutative ring R by an ideal I : Ideal R

@[reducible, inline]

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

The quotient R/I of a ring R by an ideal I.

The ideal quotient of I is defined to equal the quotient of I as an R-submodule of R. This definition uses abbrev so that typeclass instances can be shared between Ideal.Quotient I and Submodule.Quotient I.

Equations

• Ideal.instHasQuotient = Submodule.hasQuotient

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

Equations

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

def Ideal.Quotient.ringCon {R : Type u} [CommRing R] (I : Ideal R) : 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.commRing {R : Type u} [CommRing R] (I : Ideal R) : CommRing (R ⧸ I)

Equations

• Ideal.Quotient.commRing I = inferInstanceAs (CommRing (Ideal.Quotient.ringCon I).Quotient)

@[instance 100]

instance Ideal.Quotient.isScalarTower_right {R : Type u} [CommRing R] {I : Ideal R} {α : Type u_1} [SMul α R] [IsScalarTower α R R] : IsScalarTower α (R ⧸ I) (R ⧸ I)

Equations

• ⋯ = ⋯

instance Ideal.Quotient.smulCommClass {R : Type u} [CommRing R] {I : Ideal R} {α : Type u_1} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] : SMulCommClass α (R ⧸ I) (R ⧸ I)

Equations

• ⋯ = ⋯

instance Ideal.Quotient.smulCommClass' {R : Type u} [CommRing R] {I : Ideal R} {α : Type u_1} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] : SMulCommClass (R ⧸ I) α (R ⧸ I)

Equations

• ⋯ = ⋯

def Ideal.Quotient.mk {R : Type u} [CommRing R] (I : Ideal R) : 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} [CommRing R] {I : Ideal R} : Coe R (R ⧸ I)

Equations

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

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

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

instance Ideal.Quotient.inhabited {R : Type u} [CommRing R] {I : Ideal R} : Inhabited (R ⧸ I)

Equations

• Ideal.Quotient.inhabited = { default := (Ideal.Quotient.mk I) 37 }

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

@[simp]

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

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

theorem Ideal.Quotient.eq_zero_iff_dvd {R : Type u} [CommRing R] (x : R) (y : R) : (Ideal.Quotient.mk (Ideal.span {x})) y = 0 ↔ x ∣ y

@[simp]

theorem Ideal.Quotient.mk_singleton_self {R : Type u} [CommRing R] (x : R) : (Ideal.Quotient.mk (Ideal.span {x})) x = 0

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

theorem Ideal.Quotient.zero_eq_one_iff {R : Type u} [CommRing R] {I : Ideal R} : 0 = 1 ↔ I = ⊤

theorem Ideal.Quotient.zero_ne_one_iff {R : Type u} [CommRing R] {I : Ideal R} : 0 ≠ 1 ↔ I ≠ ⊤

theorem Ideal.Quotient.nontrivial {R : Type u} [CommRing R] {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I)

theorem Ideal.Quotient.subsingleton_iff {R : Type u} [CommRing R] {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤

instance Ideal.Quotient.instUniqueQuotientTop {R : Type u} [CommRing R] : Unique (R ⧸ ⊤)

Equations

• Ideal.Quotient.instUniqueQuotientTop = { default := 0, uniq := ⋯ }

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

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

Equations

• ⋯ = ⋯

theorem Ideal.Quotient.quotient_ring_saturate {R : Type u} [CommRing R] (I : Ideal R) (s : Set R) : ⇑(Ideal.Quotient.mk I) ⁻¹' (⇑(Ideal.Quotient.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.

instance Ideal.Quotient.noZeroDivisors {R : Type u} [CommRing R] (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I)

Equations

• ⋯ = ⋯

instance Ideal.Quotient.isDomain {R : Type u} [CommRing R] (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I)

Equations

• ⋯ = ⋯

theorem Ideal.Quotient.isDomain_iff_prime {R : Type u} [CommRing R] (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime

theorem Ideal.Quotient.exists_inv {R : Type u} [CommRing R] {I : Ideal R} [hI : I.IsMaximal] {a : R ⧸ I} : a ≠ 0 → ∃ (b : R ⧸ I), a * b = 1

@[reducible, inline]

noncomputable abbrev Ideal.Quotient.groupWithZero {R : Type u} [CommRing R] (I : Ideal R) [hI : I.IsMaximal] : GroupWithZero (R ⧸ I)

The quotient by a maximal ideal is a group with zero. This is a def rather than instance, since users will have computable inverses in some applications.

See note [reducible non-instances].

Equations

• Ideal.Quotient.groupWithZero I = GroupWithZero.mk ⋯ zpowRec ⋯ ⋯ ⋯ ⋯ ⋯

@[reducible, inline]

noncomputable abbrev Ideal.Quotient.field {R : Type u} [CommRing R] (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I)

The quotient by a maximal ideal is a field. This is a def rather than instance, since users will have computable inverses (and qsmul, ratCast) in some applications.

See note [reducible non-instances].

Equations

• Ideal.Quotient.field I = Field.mk ⋯ GroupWithZero.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (fun (q : ℚ≥0) => HMul.hMul ↑q) ⋯ ⋯ (fun (q : ℚ) => HMul.hMul ↑q) ⋯

theorem Ideal.Quotient.maximal_of_isField {R : Type u} [CommRing R] (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal

If the quotient by an ideal is a field, then the ideal is maximal.

theorem Ideal.Quotient.maximal_ideal_iff_isField_quotient {R : Type u} [CommRing R] (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I)

The quotient of a ring by an ideal is a field iff the ideal is maximal.

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

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

def Ideal.Quotient.factor {R : Type u} [CommRing R] (S : Ideal R) (T : Ideal R) (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

Equations

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

@[simp]

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

@[simp]

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

def Ideal.quotEquivOfEq {R : Type u_1} [CommRing R] {I : Ideal R} {J : Ideal R} (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_1} [CommRing R] {I : Ideal R} {J : Ideal R} (h : I = J) (x : R) : (Ideal.quotEquivOfEq h) ((Ideal.Quotient.mk I) x) = (Ideal.Quotient.mk J) x

@[simp]

theorem Ideal.quotEquivOfEq_symm {R : Type u_1} [CommRing R] {I : Ideal R} {J : Ideal R} (h : I = J) : (Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq ⋯

instance Ideal.modulePi {R : Type u} [CommRing R] (I : Ideal R) (ι : Type v) : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι)

R^n/I^n is a R/I-module.

Equations

• I.modulePi ι = Module.mk ⋯ ⋯

noncomputable def Ideal.piQuotEquiv {R : Type u} [CommRing R] (I : Ideal R) (ι : Type v) : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → R ⧸ I

R^n/I^n is isomorphic to (R/I)^n as an R/I-module.

Equations

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

theorem Ideal.map_pi {R : Type u} [CommRing R] (I : Ideal R) {ι : Type u_1} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ (i : ι), x i ∈ I) (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I

If f : R^n → R^m is an R-linear map and I ⊆ R is an ideal, then the image of I^n is contained in I^m.