# 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} [] :

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} [] (I : ) :
One (R I)
Equations
• = { one := }
def Ideal.Quotient.ringCon {R : Type u} [] (I : ) :

On Ideals, Submodule.quotientRel is a ring congruence.

Equations
• = let __src := ; { toSetoid := __src.toSetoid, mul' := , add' := }
Instances For
instance Ideal.Quotient.commRing {R : Type u} [] (I : ) :
Equations
@[instance 100]
instance Ideal.Quotient.isScalarTower_right {R : Type u} [] {I : } {α : Type u_1} [SMul α R] [] :
IsScalarTower α (R I) (R I)
Equations
• =
instance Ideal.Quotient.smulCommClass {R : Type u} [] {I : } {α : Type u_1} [SMul α R] [] [] :
SMulCommClass α (R I) (R I)
Equations
• =
instance Ideal.Quotient.smulCommClass' {R : Type u} [] {I : } {α : Type u_1} [SMul α R] [] [] :
SMulCommClass (R I) α (R I)
Equations
• =
def Ideal.Quotient.mk {R : Type u} [] (I : ) :
R →+* R I

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

Equations
• = { toFun := fun (a : R) => , map_one' := , map_mul' := , map_zero' := , map_add' := }
Instances For
instance Ideal.Quotient.instCoeQuotient {R : Type u} [] {I : } :
Coe R (R I)
Equations
• Ideal.Quotient.instCoeQuotient = { coe := }
theorem Ideal.Quotient.ringHom_ext {R : Type u} [] {I : } {S : Type v} [] ⦃f : R I →+* S ⦃g : R I →+* S (h : f.comp = g.comp ) :
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} [] {I : } :
Equations
• Ideal.Quotient.inhabited = { default := 37 }
theorem Ideal.Quotient.eq {R : Type u} [] {I : } {x : R} {y : R} :
x = y x - y I
@[simp]
theorem Ideal.Quotient.mk_eq_mk {R : Type u} [] {I : } (x : R) :
theorem Ideal.Quotient.eq_zero_iff_mem {R : Type u} [] {a : R} {I : } :
a = 0 a I
theorem Ideal.Quotient.eq_zero_iff_dvd {R : Type u} [] (x : R) (y : R) :
() y = 0 x y
@[simp]
theorem Ideal.Quotient.mk_singleton_self {R : Type u} [] (x : R) :
() x = 0
theorem Ideal.Quotient.mk_eq_mk_iff_sub_mem {R : Type u} [] {I : } (x : R) (y : R) :
x = y x - y I
theorem Ideal.Quotient.zero_eq_one_iff {R : Type u} [] {I : } :
0 = 1 I =
theorem Ideal.Quotient.zero_ne_one_iff {R : Type u} [] {I : } :
0 1 I
theorem Ideal.Quotient.nontrivial {R : Type u} [] {I : } (hI : I ) :
Equations
• Ideal.Quotient.instUniqueQuotientTop = { default := 0, uniq := }
theorem Ideal.Quotient.mk_surjective {R : Type u} [] {I : } :
Equations
• =
theorem Ideal.Quotient.quotient_ring_saturate {R : Type u} [] (I : ) (s : Set R) :
⁻¹' ( '' 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} [] (I : ) [hI : I.IsPrime] :
Equations
• =
instance Ideal.Quotient.isDomain {R : Type u} [] (I : ) [hI : I.IsPrime] :
Equations
• =
theorem Ideal.Quotient.isDomain_iff_prime {R : Type u} [] (I : ) :
IsDomain (R I) I.IsPrime
theorem Ideal.Quotient.exists_inv {R : Type u} [] {I : } [hI : I.IsMaximal] {a : R I} :
a 0∃ (b : R I), a * b = 1
@[reducible, inline]
noncomputable abbrev Ideal.Quotient.groupWithZero {R : Type u} [] (I : ) [hI : I.IsMaximal] :

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
Instances For
@[reducible, inline]
noncomputable abbrev Ideal.Quotient.field {R : Type u} [] (I : ) [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
• One or more equations did not get rendered due to their size.
Instances For
theorem Ideal.Quotient.maximal_of_isField {R : Type u} [] (I : ) (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} [] (I : ) :
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} [] {S : Type v} [] (I : ) (f : R →+* S) (H : aI, 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
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem Ideal.Quotient.lift_mk {R : Type u} [] {a : R} {S : Type v} [] (I : ) (f : R →+* S) (H : aI, f a = 0) :
() ( a) = f a
theorem Ideal.Quotient.lift_surjective_of_surjective {R : Type u} [] {S : Type v} [] (I : ) {f : R →+* S} (H : aI, f a = 0) (hf : ) :
def Ideal.Quotient.factor {R : Type u} [] (S : ) (T : ) (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
Instances For
@[simp]
theorem Ideal.Quotient.factor_mk {R : Type u} [] (S : ) (T : ) (H : S T) (x : R) :
() ( x) = x
@[simp]
theorem Ideal.Quotient.factor_comp_mk {R : Type u} [] (S : ) (T : ) (H : S T) :
().comp =
def Ideal.quotEquivOfEq {R : Type u_1} [] {I : } {J : } (h : I = J) :
R I ≃+* R J

Quotienting by equal ideals gives equivalent rings.

See also Submodule.quotEquivOfEq and Ideal.quotientEquivAlgOfEq.

Equations
• = let __src := ; { toFun := (__src).toFun, invFun := __src.invFun, left_inv := , right_inv := , map_mul' := , map_add' := }
Instances For
@[simp]
theorem Ideal.quotEquivOfEq_mk {R : Type u_1} [] {I : } {J : } (h : I = J) (x : R) :
( x) = x
@[simp]
theorem Ideal.quotEquivOfEq_symm {R : Type u_1} [] {I : } {J : } (h : I = J) :
.symm =
instance Ideal.modulePi {R : Type u} [] (I : ) (ι : Type v) :
Module (R I) ((ιR) I.pi ι)

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

Equations
• I.modulePi ι =
noncomputable def Ideal.piQuotEquiv {R : Type u} [] (I : ) (ι : Type v) :
((ιR) I.pi ι) ≃ₗ[R I] ιR 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.
Instances For
theorem Ideal.map_pi {R : Type u} [] (I : ) {ι : Type u_1} [] {ι' : Type w} (x : ιR) (hi : ∀ (i : ι), x i I) (f : (ιR) →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.