mathlib3 documentation

algebra.ring.equiv

(Semi)ring equivs #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file we define extension of equiv called ring_equiv, which is a datatype representing an isomorphism of semirings, rings, division_rings, or fields. We also introduce the corresponding group of automorphisms ring_aut.

Notations #

The extended equiv have coercions to functions, and the coercion is the canonical notation when treating the isomorphism as maps.

Implementation notes #

The fields for ring_equiv now avoid the unbundled is_mul_hom and is_add_hom, as these are deprecated.

Definition of multiplication in the groups of automorphisms agrees with function composition, multiplication in equiv.perm, and multiplication in category_theory.End, not with category_theory.comp.

Tags #

equiv, mul_equiv, add_equiv, ring_equiv, mul_aut, add_aut, ring_aut

source
structure ring_equiv (R : Type u_7) (S : Type u_8) [has_mul R] [has_add R] [has_mul S] [has_add S] :
Type (max u_7 u_8)

An equivalence between two (non-unital non-associative semi)rings that preserves the algebraic structure.

Instances for ring_equiv
source
def ring_equiv.to_mul_equiv {R : Type u_7} {S : Type u_8} [has_mul R] [has_add R] [has_mul S] [has_add S] (self : R ≃+* S) :
R ≃* S

The equivalence of multiplicative monoids underlying an equivalence of (semi)rings.

source
def ring_equiv.to_add_equiv {R : Type u_7} {S : Type u_8} [has_mul R] [has_add R] [has_mul S] [has_add S] (self : R ≃+* S) :
R ≃+ S

The equivalence of additive monoids underlying an equivalence of (semi)rings.

source
def ring_equiv.to_equiv {R : Type u_7} {S : Type u_8} [has_mul R] [has_add R] [has_mul S] [has_add S] (self : R ≃+* S) :
R ≃ S

The "plain" equivalence of types underlying an equivalence of (semi)rings.

source
@[class]
structure ring_equiv_class (F : Type u_7) (R : out_param (Type u_8)) (S : out_param (Type u_9)) [has_mul R] [has_add R] [has_mul S] [has_add S] :
Type (max u_7 u_8 u_9)

ring_equiv_class F R S states that F is a type of ring structure preserving equivalences. You should extend this class when you extend ring_equiv.

Instances of this typeclass
Instances of other typeclasses for ring_equiv_class
  • ring_equiv_class.has_sizeof_inst
source
@[instance]
def ring_equiv_class.to_mul_equiv_class (F : Type u_7) (R : out_param (Type u_8)) (S : out_param (Type u_9)) [has_mul R] [has_add R] [has_mul S] [has_add S] [self : ring_equiv_class F R S] :
mul_equiv_class F R S
source
@[protected, instance]
def ring_equiv_class.to_add_equiv_class (F : Type u_1) (R : Type u_2) (S : Type u_3) [has_mul R] [has_add R] [has_mul S] [has_add S] [h : ring_equiv_class F R S] :
add_equiv_class F R S
Equations
source
@[protected, instance]
def ring_equiv_class.to_ring_hom_class (F : Type u_1) (R : Type u_2) (S : Type u_3) [non_assoc_semiring R] [non_assoc_semiring S] [h : ring_equiv_class F R S] :
ring_hom_class F R S
Equations
source
@[protected, instance]
def ring_equiv_class.to_non_unital_ring_hom_class (F : Type u_1) (R : Type u_2) (S : Type u_3) [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [h : ring_equiv_class F R S] :
non_unital_ring_hom_class F R S
Equations
source
@[protected, instance]
def ring_equiv.has_coe_t {F : Type u_1} {α : Type u_2} {β : Type u_3} [has_mul α] [has_add α] [has_mul β] [has_add β] [ring_equiv_class F α β] :
has_coe_t F (α ≃+* β)
Equations
source
@[protected, instance]
def ring_equiv.ring_equiv_class {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] :
ring_equiv_class (R ≃+* S) R S
Equations
source
@[protected, instance]
def ring_equiv.has_coe_to_fun {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] :
has_coe_to_fun (R ≃+* S) (λ (_x : R ≃+* S), R → S)
Equations
source
@[simp]
theorem ring_equiv.to_equiv_eq_coe {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
f.to_equiv = ↑f
source
@[simp]
theorem ring_equiv.to_fun_eq_coe {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
f.to_fun = ⇑f
source
@[simp]
theorem ring_equiv.coe_to_equiv {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
⇑↑f = ⇑f
source
@[protected]
theorem ring_equiv.map_mul {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x y : R) :
⇑e (x * y) = ⇑e x * ⇑e y

A ring isomorphism preserves multiplication.

source
@[protected]
theorem ring_equiv.map_add {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x y : R) :
⇑e (x + y) = ⇑e x + ⇑e y

A ring isomorphism preserves addition.

source
@[ext]
theorem ring_equiv.ext {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] {f g : R ≃+* S} (h : ∀ (x : R), ⇑f x = ⇑g x) :
f = g

Two ring isomorphisms agree if they are defined by the same underlying function.

source
@[simp]
theorem ring_equiv.coe_mk {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R → S) (e' : S → R) (h₁ : function.left_inverse e' e) (h₂ : function.right_inverse e' e) (h₃ : ∀ (x y : R), e (x * y) = e x * e y) (h₄ : ∀ (x y : R), e (x + y) = e x + e y) :
⇑{to_fun := e, inv_fun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄} = e
source
@[simp]
theorem ring_equiv.mk_coe {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (e' : S → R) (h₁ : function.left_inverse e' ⇑e) (h₂ : function.right_inverse e' ⇑e) (h₃ : ∀ (x y : R), ⇑e (x * y) = ⇑e x * ⇑e y) (h₄ : ∀ (x y : R), ⇑e (x + y) = ⇑e x + ⇑e y) :
{to_fun := ⇑e, inv_fun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄} = e
source
@[protected]
theorem ring_equiv.congr_arg {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] {f : R ≃+* S} {x x' : R} :
x = x' → ⇑f x = ⇑f x'
source
@[protected]
theorem ring_equiv.congr_fun {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] {f g : R ≃+* S} (h : f = g) (x : R) :
⇑f x = ⇑g x
source
@[protected]
theorem ring_equiv.ext_iff {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] {f g : R ≃+* S} :
f = g ↔ ∀ (x : R), ⇑f x = ⇑g x
source
@[simp]
theorem ring_equiv.to_add_equiv_eq_coe {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
f.to_add_equiv = ↑f
source
@[simp]
theorem ring_equiv.to_mul_equiv_eq_coe {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
f.to_mul_equiv = ↑f
source
@[simp, norm_cast]
theorem ring_equiv.coe_to_mul_equiv {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
⇑↑f = ⇑f
source
@[simp, norm_cast]
theorem ring_equiv.coe_to_add_equiv {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
⇑↑f = ⇑f
source
def ring_equiv.ring_equiv_of_unique {M : Type u_1} {N : Type u_2} [unique M] [unique N] [has_add M] [has_mul M] [has_add N] [has_mul N] :
M ≃+* N

The ring_equiv between two semirings with a unique element.

Equations
source
@[protected, instance]
def ring_equiv.unique {M : Type u_1} {N : Type u_2} [unique M] [unique N] [has_add M] [has_mul M] [has_add N] [has_mul N] :
unique (M ≃+* N)
Equations
source
@[protected, refl]
def ring_equiv.refl (R : Type u_4) [has_mul R] [has_add R] :
R ≃+* R

The identity map is a ring isomorphism.

Equations
source
@[simp]
theorem ring_equiv.refl_apply (R : Type u_4) [has_mul R] [has_add R] (x : R) :
⇑(ring_equiv.refl R) x = x
source
@[simp]
theorem ring_equiv.coe_add_equiv_refl (R : Type u_4) [has_mul R] [has_add R] :
↑(ring_equiv.refl R) = add_equiv.refl R
source
@[simp]
theorem ring_equiv.coe_mul_equiv_refl (R : Type u_4) [has_mul R] [has_add R] :
↑(ring_equiv.refl R) = mul_equiv.refl R
source
@[protected, instance]
def ring_equiv.inhabited (R : Type u_4) [has_mul R] [has_add R] :
inhabited (R ≃+* R)
Equations
source
@[protected, symm]
def ring_equiv.symm {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
S ≃+* R

The inverse of a ring isomorphism is a ring isomorphism.

Equations
source
def ring_equiv.simps.symm_apply {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
S → R

See Note [custom simps projection]

Equations
source
@[simp]
theorem ring_equiv.inv_fun_eq_symm {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :
f.inv_fun = ⇑(f.symm)
source
@[simp]
theorem ring_equiv.symm_symm {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
e.symm.symm = e
source
@[simp]
theorem ring_equiv.coe_to_equiv_symm {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
↑(e.symm) = ↑e.symm
source
theorem ring_equiv.symm_bijective {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] :
function.bijective ring_equiv.symm
source
@[simp]
theorem ring_equiv.mk_coe' {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (f : S → R) (h₁ : function.left_inverse ⇑e f) (h₂ : function.right_inverse ⇑e f) (h₃ : ∀ (x y : S), f (x * y) = f x * f y) (h₄ : ∀ (x y : S), f (x + y) = f x + f y) :
{to_fun := f, inv_fun := ⇑e, left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄} = e.symm
source
@[simp]
theorem ring_equiv.symm_mk {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R → S) (g : S → R) (h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) (h₃ : ∀ (x y : R), f (x * y) = f x * f y) (h₄ : ∀ (x y : R), f (x + y) = f x + f y) :
{to_fun := f, inv_fun := g, left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄}.symm = {to_fun := g, inv_fun := f, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}
source
@[protected, trans]
def ring_equiv.trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
R ≃+* S'

Transitivity of ring_equiv.

Equations
source
theorem ring_equiv.trans_apply {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) :
⇑(e₁.trans e₂) a = ⇑e₂ (⇑e₁ a)
source
@[simp]
theorem ring_equiv.coe_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁
source
@[simp]
theorem ring_equiv.symm_trans_apply {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : S') :
⇑((e₁.trans e₂).symm) a = ⇑(e₁.symm) (⇑(e₂.symm) a)
source
theorem ring_equiv.symm_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂).symm = e₂.symm.trans e₁.symm
source
@[protected]
theorem ring_equiv.bijective {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
function.bijective ⇑e
source
@[protected]
theorem ring_equiv.injective {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
function.injective ⇑e
source
@[protected]
theorem ring_equiv.surjective {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :
function.surjective ⇑e
source
@[simp]
theorem ring_equiv.apply_symm_apply {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x : S) :
⇑e (⇑(e.symm) x) = x
source
@[simp]
theorem ring_equiv.symm_apply_apply {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x : R) :
⇑(e.symm) (⇑e x) = x
source
theorem ring_equiv.image_eq_preimage {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (s : set R) :
⇑e '' s = ⇑(e.symm) ⁻¹' s
source
@[simp]
theorem ring_equiv.coe_mul_equiv_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
↑(e₁.trans e₂) = ↑e₁.trans ↑e₂
source
@[simp]
theorem ring_equiv.coe_add_equiv_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
↑(e₁.trans e₂) = ↑e₁.trans ↑e₂
source
@[simp]
theorem ring_equiv.op_symm_apply_symm_apply {α : Type u_1} {β : Type u_2} [has_add α] [has_mul α] [has_add β] [has_mul β] (f : αᵐᵒᵖ ≃+* βᵐᵒᵖ) (ᾰ : β) :
⇑((⇑(ring_equiv.op.symm) f).symm) ᾰ = (⇑(add_equiv.mul_op.symm) f.to_add_equiv).inv_fun ᾰ
source
@[simp]
theorem ring_equiv.op_symm_apply_apply {α : Type u_1} {β : Type u_2} [has_add α] [has_mul α] [has_add β] [has_mul β] (f : αᵐᵒᵖ ≃+* βᵐᵒᵖ) (ᾰ : α) :
⇑(⇑(ring_equiv.op.symm) f) ᾰ = (⇑(add_equiv.mul_op.symm) f.to_add_equiv).to_fun ᾰ
source
@[simp]
theorem ring_equiv.op_apply_symm_apply {α : Type u_1} {β : Type u_2} [has_add α] [has_mul α] [has_add β] [has_mul β] (f : α ≃+* β) (ᾰ : βᵐᵒᵖ) :
⇑((⇑ring_equiv.op f).symm) ᾰ = (⇑add_equiv.mul_op f.to_add_equiv).inv_fun ᾰ
source
@[protected]
def ring_equiv.op {α : Type u_1} {β : Type u_2} [has_add α] [has_mul α] [has_add β] [has_mul β] :
α ≃+* β ≃ (αᵐᵒᵖ ≃+* βᵐᵒᵖ)

A ring iso α ≃+* β can equivalently be viewed as a ring iso αᵐᵒᵖ ≃+* βᵐᵒᵖ.

Equations
source
@[simp]
theorem ring_equiv.op_apply_apply {α : Type u_1} {β : Type u_2} [has_add α] [has_mul α] [has_add β] [has_mul β] (f : α ≃+* β) (ᾰ : αᵐᵒᵖ) :
⇑(⇑ring_equiv.op f) ᾰ = (⇑add_equiv.mul_op f.to_add_equiv).to_fun ᾰ
source
@[protected, simp]
def ring_equiv.unop {α : Type u_1} {β : Type u_2} [has_add α] [has_mul α] [has_add β] [has_mul β] :
αᵐᵒᵖ ≃+* βᵐᵒᵖ ≃ (α ≃+* β)

The 'unopposite' of a ring iso αᵐᵒᵖ ≃+* βᵐᵒᵖ. Inverse to ring_equiv.op.

Equations
source
def ring_equiv.to_opposite (R : Type u_4) [non_unital_comm_semiring R] :
R ≃+* Rᵐᵒᵖ

A non-unital commutative ring is isomorphic to its opposite.

Equations
source
@[simp]
theorem ring_equiv.to_opposite_apply (R : Type u_4) [non_unital_comm_semiring R] (r : R) :
⇑(ring_equiv.to_opposite R) r = mul_opposite.op r
source
@[simp]
theorem ring_equiv.to_opposite_symm_apply (R : Type u_4) [non_unital_comm_semiring R] (r : Rᵐᵒᵖ) :
⇑((ring_equiv.to_opposite R).symm) r = mul_opposite.unop r
source
@[protected]
theorem ring_equiv.map_zero {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) :
⇑f 0 = 0

A ring isomorphism sends zero to zero.

source
@[protected]
theorem ring_equiv.map_eq_zero_iff {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) {x : R} :
⇑f x = 0 ↔ x = 0
source
theorem ring_equiv.map_ne_zero_iff {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) {x : R} :
⇑f x ≠ 0 ↔ x ≠ 0
source
noncomputable def ring_equiv.of_bijective {F : Type u_1} {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S] (f : F) (hf : function.bijective ⇑f) :
R ≃+* S

Produce a ring isomorphism from a bijective ring homomorphism.

Equations
source
@[simp]
theorem ring_equiv.coe_of_bijective {F : Type u_1} {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S] (f : F) (hf : function.bijective ⇑f) :
⇑(ring_equiv.of_bijective f hf) = ⇑f
source
theorem ring_equiv.of_bijective_apply {F : Type u_1} {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S] (f : F) (hf : function.bijective ⇑f) (x : R) :
⇑(ring_equiv.of_bijective f hf) x = ⇑f x
source
@[simp]
theorem ring_equiv.Pi_congr_right_apply {ι : Type u_1} {R : ι → Type u_2} {S : ι → Type u_3} [Π (i : ι), non_unital_non_assoc_semiring (R i)] [Π (i : ι), non_unital_non_assoc_semiring (S i)] (e : Π (i : ι), R i ≃+* S i) (x : Π (i : ι), R i) (j : ι) :
⇑(ring_equiv.Pi_congr_right e) x j = ⇑(e j) (x j)
source
def ring_equiv.Pi_congr_right {ι : Type u_1} {R : ι → Type u_2} {S : ι → Type u_3} [Π (i : ι), non_unital_non_assoc_semiring (R i)] [Π (i : ι), non_unital_non_assoc_semiring (S i)] (e : Π (i : ι), R i ≃+* S i) :
(Π (i : ι), R i) ≃+* Π (i : ι), S i

A family of ring isomorphisms Π j, (R j ≃+* S j) generates a ring isomorphisms between Π j, R j and Π j, S j.

This is the ring_equiv version of equiv.Pi_congr_right, and the dependent version of ring_equiv.arrow_congr.

Equations
source
@[simp]
theorem ring_equiv.Pi_congr_right_refl {ι : Type u_1} {R : ι → Type u_2} [Π (i : ι), non_unital_non_assoc_semiring (R i)] :
ring_equiv.Pi_congr_right (λ (i : ι), ring_equiv.refl (R i)) = ring_equiv.refl (Π (i : ι), R i)
source
@[simp]
theorem ring_equiv.Pi_congr_right_symm {ι : Type u_1} {R : ι → Type u_2} {S : ι → Type u_3} [Π (i : ι), non_unital_non_assoc_semiring (R i)] [Π (i : ι), non_unital_non_assoc_semiring (S i)] (e : Π (i : ι), R i ≃+* S i) :
(ring_equiv.Pi_congr_right e).symm = ring_equiv.Pi_congr_right (λ (i : ι), (e i).symm)
source
@[simp]
theorem ring_equiv.Pi_congr_right_trans {ι : Type u_1} {R : ι → Type u_2} {S : ι → Type u_3} {T : ι → Type u_4} [Π (i : ι), non_unital_non_assoc_semiring (R i)] [Π (i : ι), non_unital_non_assoc_semiring (S i)] [Π (i : ι), non_unital_non_assoc_semiring (T i)] (e : Π (i : ι), R i ≃+* S i) (f : Π (i : ι), S i ≃+* T i) :
(ring_equiv.Pi_congr_right e).trans (ring_equiv.Pi_congr_right f) = ring_equiv.Pi_congr_right (λ (i : ι), (e i).trans (f i))
source
@[protected]
theorem ring_equiv.map_one {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
⇑f 1 = 1

A ring isomorphism sends one to one.

source
@[protected]
theorem ring_equiv.map_eq_one_iff {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) {x : R} :
⇑f x = 1 ↔ x = 1
source
theorem ring_equiv.map_ne_one_iff {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) {x : R} :
⇑f x ≠ 1 ↔ x ≠ 1
source
theorem ring_equiv.coe_monoid_hom_refl {R : Type u_4} [non_assoc_semiring R] :
↑(ring_equiv.refl R) = monoid_hom.id R
source
@[simp]
theorem ring_equiv.coe_add_monoid_hom_refl {R : Type u_4} [non_assoc_semiring R] :
↑(ring_equiv.refl R) = add_monoid_hom.id R

ring_equiv.coe_mul_equiv_refl and ring_equiv.coe_add_equiv_refl are proved above in higher generality

source
@[simp]
theorem ring_equiv.coe_ring_hom_refl {R : Type u_4} [non_assoc_semiring R] :
↑(ring_equiv.refl R) = ↑(ring_hom.id R)
source
@[simp]
theorem ring_equiv.coe_monoid_hom_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
↑(e₁.trans e₂) = ↑e₂.comp ↑e₁
source
@[simp]
theorem ring_equiv.coe_add_monoid_hom_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
↑(e₁.trans e₂) = ↑e₂.comp ↑e₁

ring_equiv.coe_mul_equiv_trans and ring_equiv.coe_add_equiv_trans are proved above in higher generality

source
@[simp]
theorem ring_equiv.coe_ring_hom_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
↑(e₁.trans e₂) = ↑e₂.comp ↑e₁
source
@[simp]
theorem ring_equiv.comp_symm {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
↑e.comp ↑(e.symm) = ring_hom.id S
source
@[simp]
theorem ring_equiv.symm_comp {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
↑(e.symm).comp ↑e = ring_hom.id R
source
@[protected]
theorem ring_equiv.map_neg {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_ring R] [non_unital_non_assoc_ring S] (f : R ≃+* S) (x : R) :
⇑f (-x) = -⇑f x
source
@[protected]
theorem ring_equiv.map_sub {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_ring R] [non_unital_non_assoc_ring S] (f : R ≃+* S) (x y : R) :
⇑f (x - y) = ⇑f x - ⇑f y
source
@[simp]
theorem ring_equiv.map_neg_one {R : Type u_4} {S : Type u_5} [non_assoc_ring R] [non_assoc_ring S] (f : R ≃+* S) :
⇑f (-1) = -1
source
theorem ring_equiv.map_eq_neg_one_iff {R : Type u_4} {S : Type u_5} [non_assoc_ring R] [non_assoc_ring S] (f : R ≃+* S) {x : R} :
⇑f x = -1 ↔ x = -1
source
def ring_equiv.to_non_unital_ring_hom {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (e : R ≃+* S) :
R →ₙ+* S

Reinterpret a ring equivalence as a non-unital ring homomorphism.

Equations
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theorem ring_equiv.to_non_unital_ring_hom_injective {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] :
function.injective ring_equiv.to_non_unital_ring_hom
source
@[protected, instance]
def ring_equiv.has_coe_to_non_unital_ring_hom {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] :
has_coe (R ≃+* S) (R →ₙ+* S)
Equations
source
theorem ring_equiv.to_non_unital_ring_hom_eq_coe {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) :
f.to_non_unital_ring_hom = ↑f
source
@[simp, norm_cast]
theorem ring_equiv.coe_to_non_unital_ring_hom {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) :
⇑↑f = ⇑f
source
theorem ring_equiv.coe_non_unital_ring_hom_inj_iff {R : Type u_1} {S : Type u_2} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f g : R ≃+* S) :
f = g ↔ ↑f = ↑g
source
@[simp]
theorem ring_equiv.to_non_unital_ring_hom_refl {R : Type u_4} [non_unital_non_assoc_semiring R] :
(ring_equiv.refl R).to_non_unital_ring_hom = non_unital_ring_hom.id R
source
@[simp]
theorem ring_equiv.to_non_unital_ring_hom_apply_symm_to_non_unital_ring_hom_apply {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (e : R ≃+* S) (y : S) :
⇑(e.to_non_unital_ring_hom) (⇑(e.symm.to_non_unital_ring_hom) y) = y
source
@[simp]
theorem ring_equiv.symm_to_non_unital_ring_hom_apply_to_non_unital_ring_hom_apply {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (e : R ≃+* S) (x : R) :
⇑(e.symm.to_non_unital_ring_hom) (⇑(e.to_non_unital_ring_hom) x) = x
source
@[simp]
theorem ring_equiv.to_non_unital_ring_hom_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂).to_non_unital_ring_hom = e₂.to_non_unital_ring_hom.comp e₁.to_non_unital_ring_hom
source
@[simp]
theorem ring_equiv.to_non_unital_ring_hom_comp_symm_to_non_unital_ring_hom {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (e : R ≃+* S) :
e.to_non_unital_ring_hom.comp e.symm.to_non_unital_ring_hom = non_unital_ring_hom.id S
source
@[simp]
theorem ring_equiv.symm_to_non_unital_ring_hom_comp_to_non_unital_ring_hom {R : Type u_4} {S : Type u_5} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (e : R ≃+* S) :
e.symm.to_non_unital_ring_hom.comp e.to_non_unital_ring_hom = non_unital_ring_hom.id R
source
def ring_equiv.to_ring_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
R →+* S

Reinterpret a ring equivalence as a ring homomorphism.

Equations
source
theorem ring_equiv.to_ring_hom_injective {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] :
function.injective ring_equiv.to_ring_hom
source
@[protected, instance]
def ring_equiv.has_coe_to_ring_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] :
has_coe (R ≃+* S) (R →+* S)
Equations
source
theorem ring_equiv.to_ring_hom_eq_coe {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
f.to_ring_hom = ↑f
source
@[simp, norm_cast]
theorem ring_equiv.coe_to_ring_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
⇑↑f = ⇑f
source
theorem ring_equiv.coe_ring_hom_inj_iff {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f g : R ≃+* S) :
f = g ↔ ↑f = ↑g
source
@[simp, norm_cast]
theorem ring_equiv.to_non_unital_ring_hom_commutes {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
↑↑f = ↑f

The two paths coercion can take to a non_unital_ring_hom are equivalent

source
@[reducible]
def ring_equiv.to_monoid_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
R →* S

Reinterpret a ring equivalence as a monoid homomorphism.

source
@[reducible]
def ring_equiv.to_add_monoid_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
R →+ S

Reinterpret a ring equivalence as an add_monoid homomorphism.

source
theorem ring_equiv.to_add_monoid_hom_commutes {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
↑f.to_add_monoid_hom = ↑f.to_add_monoid_hom

The two paths coercion can take to an add_monoid_hom are equivalent

source
theorem ring_equiv.to_monoid_hom_commutes {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
↑f.to_monoid_hom = ↑f.to_monoid_hom

The two paths coercion can take to an monoid_hom are equivalent

source
theorem ring_equiv.to_equiv_commutes {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :
↑f.to_equiv = ↑f.to_equiv

The two paths coercion can take to an equiv are equivalent

source
@[simp]
theorem ring_equiv.to_ring_hom_refl {R : Type u_4} [non_assoc_semiring R] :
(ring_equiv.refl R).to_ring_hom = ring_hom.id R
source
@[simp]
theorem ring_equiv.to_monoid_hom_refl {R : Type u_4} [non_assoc_semiring R] :
(ring_equiv.refl R).to_monoid_hom = monoid_hom.id R
source
@[simp]
theorem ring_equiv.to_add_monoid_hom_refl {R : Type u_4} [non_assoc_semiring R] :
(ring_equiv.refl R).to_add_monoid_hom = add_monoid_hom.id R
source
@[simp]
theorem ring_equiv.to_ring_hom_apply_symm_to_ring_hom_apply {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (y : S) :
⇑(e.to_ring_hom) (⇑(e.symm.to_ring_hom) y) = y
source
@[simp]
theorem ring_equiv.symm_to_ring_hom_apply_to_ring_hom_apply {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (x : R) :
⇑(e.symm.to_ring_hom) (⇑(e.to_ring_hom) x) = x
source
@[simp]
theorem ring_equiv.to_ring_hom_trans {R : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂).to_ring_hom = e₂.to_ring_hom.comp e₁.to_ring_hom
source
@[simp]
theorem ring_equiv.to_ring_hom_comp_symm_to_ring_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
e.to_ring_hom.comp e.symm.to_ring_hom = ring_hom.id S
source
@[simp]
theorem ring_equiv.symm_to_ring_hom_comp_to_ring_hom {R : Type u_4} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :
e.symm.to_ring_hom.comp e.to_ring_hom = ring_hom.id R
source
def ring_equiv.of_hom_inv' {R : Type u_1} {S : Type u_2} {F : Type u_3} {G : Type u_4} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S] [non_unital_ring_hom_class G S R] (hom : F) (inv : G) (hom_inv_id : ↑inv.comp ↑hom = non_unital_ring_hom.id R) (inv_hom_id : ↑hom.comp ↑inv = non_unital_ring_hom.id S) :
R ≃+* S

Construct an equivalence of rings from homomorphisms in both directions, which are inverses.

Equations
source
@[simp]
theorem ring_equiv.of_hom_inv'_apply {R : Type u_1} {S : Type u_2} {F : Type u_3} {G : Type u_4} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S] [non_unital_ring_hom_class G S R] (hom : F) (inv : G) (hom_inv_id : ↑inv.comp ↑hom = non_unital_ring_hom.id R) (inv_hom_id : ↑hom.comp ↑inv = non_unital_ring_hom.id S) (a : R) :
⇑(ring_equiv.of_hom_inv' hom inv hom_inv_id inv_hom_id) a = ⇑hom a
source
@[simp]
theorem ring_equiv.of_hom_inv'_symm_apply {R : Type u_1} {S : Type u_2} {F : Type u_3} {G : Type u_4} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S] [non_unital_ring_hom_class G S R] (hom : F) (inv : G) (hom_inv_id : ↑inv.comp ↑hom = non_unital_ring_hom.id R) (inv_hom_id : ↑hom.comp ↑inv = non_unital_ring_hom.id S) (a : S) :
⇑((ring_equiv.of_hom_inv' hom inv hom_inv_id inv_hom_id).symm) a = ⇑inv a
source
@[simp]
theorem ring_equiv.of_hom_inv_symm_apply {R : Type u_1} {S : Type u_2} {F : Type u_3} {G : Type u_4} [non_assoc_semiring R] [non_assoc_semiring S] [ring_hom_class F R S] [ring_hom_class G S R] (hom : F) (inv : G) (hom_inv_id : ↑inv.comp ↑hom = ring_hom.id R) (inv_hom_id : ↑hom.comp ↑inv = ring_hom.id S) (a : S) :
⇑((ring_equiv.of_hom_inv hom inv hom_inv_id inv_hom_id).symm) a = ⇑inv a
source
def ring_equiv.of_hom_inv {R : Type u_1} {S : Type u_2} {F : Type u_3} {G : Type u_4} [non_assoc_semiring R] [non_assoc_semiring S] [ring_hom_class F R S] [ring_hom_class G S R] (hom : F) (inv : G) (hom_inv_id : ↑inv.comp ↑hom = ring_hom.id R) (inv_hom_id : ↑hom.comp ↑inv = ring_hom.id S) :
R ≃+* S

Construct an equivalence of rings from unital homomorphisms in both directions, which are inverses.

Equations
source
@[simp]
theorem ring_equiv.of_hom_inv_apply {R : Type u_1} {S : Type u_2} {F : Type u_3} {G : Type u_4} [non_assoc_semiring R] [non_assoc_semiring S] [ring_hom_class F R S] [ring_hom_class G S R] (hom : F) (inv : G) (hom_inv_id : ↑inv.comp ↑hom = ring_hom.id R) (inv_hom_id : ↑hom.comp ↑inv = ring_hom.id S) (a : R) :
⇑(ring_equiv.of_hom_inv hom inv hom_inv_id inv_hom_id) a = ⇑hom a
source
@[protected]
theorem ring_equiv.map_pow {R : Type u_4} {S : Type u_5} [semiring R] [semiring S] (f : R ≃+* S) (a : R) (n : ℕ) :
⇑f (a ^ n) = ⇑f a ^ n
source
def mul_equiv.to_ring_equiv {R : Type u_1} {S : Type u_2} {F : Type u_3} [has_add R] [has_add S] [has_mul R] [has_mul S] [mul_equiv_class F R S] (f : F) (H : ∀ (x y : R), ⇑f (x + y) = ⇑f x + ⇑f y) :
R ≃+* S

Gives a ring_equiv from an element of a mul_equiv_class preserving addition.

Equations
source
def add_equiv.to_ring_equiv {R : Type u_1} {S : Type u_2} {F : Type u_3} [has_add R] [has_add S] [has_mul R] [has_mul S] [add_equiv_class F R S] (f : F) (H : ∀ (x y : R), ⇑f (x * y) = ⇑f x * ⇑f y) :
R ≃+* S

Gives a ring_equiv from an element of an add_equiv_class preserving addition.

Equations
source
@[simp]
theorem ring_equiv.self_trans_symm {R : Type u_4} {S : Type u_5} [has_add R] [has_add S] [has_mul R] [has_mul S] (e : R ≃+* S) :
e.trans e.symm = ring_equiv.refl R
source
@[simp]
theorem ring_equiv.symm_trans_self {R : Type u_4} {S : Type u_5} [has_add R] [has_add S] [has_mul R] [has_mul S] (e : R ≃+* S) :
e.symm.trans e = ring_equiv.refl S
source
@[protected]
theorem ring_equiv.no_zero_divisors {A : Type u_1} (B : Type u_2) [ring A] [ring B] [no_zero_divisors B] (e : A ≃+* B) :
no_zero_divisors A

If two rings are isomorphic, and the second doesn't have zero divisors, then so does the first.

source
@[protected]
theorem ring_equiv.is_domain {A : Type u_1} (B : Type u_2) [ring A] [ring B] [is_domain B] (e : A ≃+* B) :
is_domain A

If two rings are isomorphic, and the second is a domain, then so is the first.