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 semirings.

Main definitions #

Ideal.Quotient.Ring: the quotient of a ring R by a two-sided ideal I : Ideal R

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@[simp]

theorem Ideal.Quotient.mk_span_range {ι : Type u_1} {R : Type u_3} [Ring R] (f : ι → R) [(span (Set.range f)).IsTwoSided] (i : ι) :

(mk (span (Set.range f))) (f i) = 0

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theorem Ideal.Quotient.zero_eq_one_iff {R : Type u_3} [Ring R] {I : Ideal R} :

0 = 1 ↔ I = ⊤

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theorem Ideal.Quotient.zero_ne_one_iff {R : Type u_3} [Ring R] {I : Ideal R} :

0 ≠ 1 ↔ I ≠ ⊤

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theorem Ideal.Quotient.subsingleton_iff {R : Type u_3} [Ring R] {I : Ideal R} :

Subsingleton (R ⧸ I) ↔ I = ⊤

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theorem Ideal.Quotient.nontrivial_iff {R : Type u_3} [Ring R] {I : Ideal R} :

Nontrivial (R ⧸ I) ↔ I ≠ ⊤

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@[deprecated Ideal.Quotient.nontrivial_iff (since := "2025-11-02")]

theorem Ideal.Quotient.nontrivial {R : Type u_3} [Ring R] {I : Ideal R} (hI : I ≠ ⊤) :

Nontrivial (R ⧸ I)

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instance Ideal.Quotient.instUniqueQuotientTop {R : Type u_3} [Ring R] :

Unique (R ⧸ ⊤)

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@[instance 100]

instance Ideal.Quotient.isScalarTower_right {R : Type u_3} [Ring R] {I : Ideal R} [I.IsTwoSided] {α : Type u_5} [SMul α R] [IsScalarTower α R R] :

IsScalarTower α (R ⧸ I) (R ⧸ I)

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instance Ideal.Quotient.smulCommClass {R : Type u_3} [Ring R] {I : Ideal R} [I.IsTwoSided] {α : Type u_5} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :

SMulCommClass α (R ⧸ I) (R ⧸ I)

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instance Ideal.Quotient.smulCommClass' {R : Type u_3} [Ring R] {I : Ideal R} [I.IsTwoSided] {α : Type u_5} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :

SMulCommClass (R ⧸ I) α (R ⧸ I)

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theorem Ideal.Quotient.eq_zero_iff_dvd {R : Type u_5} [CommRing R] (x y : R) :

(mk (span {x})) y = 0 ↔ x ∣ y

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@[simp]

theorem Ideal.Quotient.mk_singleton_self {R : Type u_3} [Ring R] (x : R) [(span {x}).IsTwoSided] :

(mk (span {x})) x = 0

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instance Ideal.Quotient.noZeroDivisors {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [hI : I.IsPrime] :

NoZeroDivisors (R ⧸ I)

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instance Ideal.Quotient.isDomain {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [hI : I.IsPrime] :

IsDomain (R ⧸ I)

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theorem Ideal.Quotient.isDomain_iff_prime {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] :

IsDomain (R ⧸ I) ↔ I.IsPrime

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theorem Ideal.Quotient.exists_inv {R : Type u_3} [Ring R] {I : Ideal R} [I.IsTwoSided] [hI : I.IsMaximal] {a : R ⧸ I} :

a ≠ 0 → ∃ (b : R ⧸ I), a * b = 1

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@[reducible, inline]

noncomputable abbrev Ideal.Quotient.groupWithZero {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [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].

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@[reducible, inline]

noncomputable abbrev Ideal.Quotient.divisionRing {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [I.IsMaximal] :

DivisionRing (R ⧸ I)

The quotient by a two-sided ideal that is maximal as a left ideal is a division ring. This is a def rather than instance, since users will have computable inverses (and qsmul, ratCast) in some applications.

See note [reducible non-instances].

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@[reducible, inline]

noncomputable abbrev Ideal.Quotient.field {R : Type u_5} [CommRing R] (I : Ideal R) [I.IsMaximal] :

Field (R ⧸ I)

The quotient of a commutative ring 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].

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theorem Ideal.Quotient.maximal_of_isField {R : Type u_5} [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.

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theorem Ideal.Quotient.maximal_ideal_iff_isField_quotient {R : Type u_5} [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.

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instance Ideal.modulePi {ι : Type u_1} {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] :

Module (R ⧸ I) ((ι → R) ⧸ pi fun (x : ι) => I)

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

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noncomputable def Ideal.piQuotEquiv (ι : Type u_1) {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] :

((ι → R) ⧸ pi fun (x : ι) => I) ≃ₗ[R ⧸ I] ι → R ⧸ I

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

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theorem Ideal.map_pi {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [Finite ι] (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.

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