mathlib3 documentation

@[protected, instance]

def subring_class.add_subgroup_class (S : Type u_1) (R : Type u) [set_like S R] [ring R] [h : subring_class S R] :

add_subgroup_class S R

theorem coe_int_mem {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) (n : ℤ) :

↑n ∈ s

@[protected, instance]

def subring_class.to_has_int_cast {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) :

has_int_cast ↥s

Equations

subring_class.to_has_int_cast s = {int_cast := λ (n : ℤ), ⟨↑n, _⟩}

@[protected, instance]

def subring_class.to_ring {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) :

ring ↥s

A subring of a ring inherits a ring structure

Equations

subring_class.to_ring s = function.injective.ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring_class.to_comm_ring {S : Type v} (s : S) {R : Type u_1} [comm_ring R] [set_like S R] [subring_class S R] :

comm_ring ↥s

A subring of a comm_ring is a comm_ring.

Equations

subring_class.to_comm_ring s = function.injective.comm_ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring_class.is_domain {S : Type v} (s : S) {R : Type u_1} [ring R] [is_domain R] [set_like S R] [subring_class S R] :

is_domain ↥s

A subring of a domain is a domain.

@[protected, instance]

def subring_class.to_ordered_ring {S : Type v} (s : S) {R : Type u_1} [ordered_ring R] [set_like S R] [subring_class S R] :

ordered_ring ↥s

A subring of an ordered_ring is an ordered_ring.

Equations

subring_class.to_ordered_ring s = function.injective.ordered_ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring_class.to_ordered_comm_ring {S : Type v} (s : S) {R : Type u_1} [ordered_comm_ring R] [set_like S R] [subring_class S R] :

ordered_comm_ring ↥s

A subring of an ordered_comm_ring is an ordered_comm_ring.

Equations

subring_class.to_ordered_comm_ring s = function.injective.ordered_comm_ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring_class.to_linear_ordered_ring {S : Type v} (s : S) {R : Type u_1} [linear_ordered_ring R] [set_like S R] [subring_class S R] :

linear_ordered_ring ↥s

A subring of a linear_ordered_ring is a linear_ordered_ring.

Equations

subring_class.to_linear_ordered_ring s = function.injective.linear_ordered_ring coe _ _ _ _ _ _ _ _ _ _ _ _ _ _

linear_ordered_comm_ring ↥s

@[protected, instance]

def subring_class.to_linear_ordered_comm_ring {S : Type v} (s : S) {R : Type u_1} [linear_ordered_comm_ring R] [set_like S R] [subring_class S R] :

A subring of a linear_ordered_comm_ring is a linear_ordered_comm_ring.

Equations

subring_class.to_linear_ordered_comm_ring s = function.injective.linear_ordered_comm_ring coe _ _ _ _ _ _ _ _ _ _ _ _ _ _

def subring_class.subtype {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) :

↥s →+* R

The natural ring hom from a subring of ring R to R.

Equations

subring_class.subtype s = {to_fun := coe coe_to_lift, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

@[simp]

theorem subring_class.coe_subtype {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) :

⇑(subring_class.subtype s) = coe

@[simp, norm_cast]

theorem subring_class.coe_nat_cast {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) (n : ℕ) :

@[simp, norm_cast]

theorem subring_class.coe_int_cast {R : Type u} {S : Type v} [ring R] [set_like S R] [hSR : subring_class S R] (s : S) (n : ℤ) :

def subring.to_add_subgroup {R : Type u} [ring R] (self : subring R) :

add_subgroup R

Reinterpret a subring as an add_subgroup.

structure subring (R : Type u) [ring R] :

Type u

carrier : set R
mul_mem' : ∀ {a b : R}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier
one_mem' : 1 ∈ self.carrier
add_mem' : ∀ {a b : R}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
zero_mem' : 0 ∈ self.carrier
neg_mem' : ∀ {x : R}, x ∈ self.carrier → -x ∈ self.carrier

subring R is the type of subrings of R. A subring of R is a subset s that is a multiplicative submonoid and an additive subgroup. Note in particular that it shares the same 0 and 1 as R.

Instances for subring

def subring.to_subsemiring {R : Type u} [ring R] (self : subring R) :

subsemiring R

Reinterpret a subring as a subsemiring.

def subring.to_submonoid {R : Type u} [ring R] (s : subring R) :

submonoid R

The underlying submonoid of a subring.

Equations

s.to_submonoid = {carrier := s.carrier, mul_mem' := _, one_mem' := _}

@[protected, instance]

def subring.set_like {R : Type u} [ring R] :

set_like (subring R) R

Equations

subring.set_like = {coe := subring.carrier _inst_1, coe_injective' := _}

@[protected, instance]

def subring.subring_class {R : Type u} [ring R] :

subring_class (subring R) R

@[simp]

theorem subring.mem_carrier {R : Type u} [ring R] {s : subring R} {x : R} :

x ∈ s.carrier ↔ x ∈ s

@[simp]

theorem subring.mem_mk {R : Type u} [ring R] {S : set R} {x : R} (h₁ : ∀ {a b : R}, a ∈ S → b ∈ S → a * b ∈ S) (h₂ : 1 ∈ S) (h₃ : ∀ {a b : R}, a ∈ S → b ∈ S → a + b ∈ S) (h₄ : 0 ∈ S) (h₅ : ∀ {x : R}, x ∈ S → -x ∈ S) :

x ∈ {carrier := S, mul_mem' := h₁, one_mem' := h₂, add_mem' := h₃, zero_mem' := h₄, neg_mem' := h₅} ↔ x ∈ S

@[simp]

theorem subring.coe_set_mk {R : Type u} [ring R] (S : set R) (h₁ : ∀ {a b : R}, a ∈ S → b ∈ S → a * b ∈ S) (h₂ : 1 ∈ S) (h₃ : ∀ {a b : R}, a ∈ S → b ∈ S → a + b ∈ S) (h₄ : 0 ∈ S) (h₅ : ∀ {x : R}, x ∈ S → -x ∈ S) :

↑{carrier := S, mul_mem' := h₁, one_mem' := h₂, add_mem' := h₃, zero_mem' := h₄, neg_mem' := h₅} = S

@[simp]

theorem subring.mk_le_mk {R : Type u} [ring R] {S S' : set R} (h₁ : ∀ {a b : R}, a ∈ S → b ∈ S → a * b ∈ S) (h₂ : 1 ∈ S) (h₃ : ∀ {a b : R}, a ∈ S → b ∈ S → a + b ∈ S) (h₄ : 0 ∈ S) (h₅ : ∀ {x : R}, x ∈ S → -x ∈ S) (h₁' : ∀ {a b : R}, a ∈ S' → b ∈ S' → a * b ∈ S') (h₂' : 1 ∈ S') (h₃' : ∀ {a b : R}, a ∈ S' → b ∈ S' → a + b ∈ S') (h₄' : 0 ∈ S') (h₅' : ∀ {x : R}, x ∈ S' → -x ∈ S') :

{carrier := S, mul_mem' := h₁, one_mem' := h₂, add_mem' := h₃, zero_mem' := h₄, neg_mem' := h₅} ≤ {carrier := S', mul_mem' := h₁', one_mem' := h₂', add_mem' := h₃', zero_mem' := h₄', neg_mem' := h₅'} ↔ S ⊆ S'

@[ext]

theorem subring.ext {R : Type u} [ring R] {S T : subring R} (h : ∀ (x : R), x ∈ S ↔ x ∈ T) :

S = T

Two subrings are equal if they have the same elements.

@[protected]

def subring.copy {R : Type u} [ring R] (S : subring R) (s : set R) (hs : s = ↑S) :

Copy of a subring with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations

S.copy s hs = {carrier := s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[simp]

theorem subring.coe_copy {R : Type u} [ring R] (S : subring R) (s : set R) (hs : s = ↑S) :

↑(S.copy s hs) = s

theorem subring.copy_eq {R : Type u} [ring R] (S : subring R) (s : set R) (hs : s = ↑S) :

S.copy s hs = S

function.injective subring.to_subsemiring

theorem subring.to_subsemiring_injective {R : Type u} [ring R] :

strict_mono subring.to_subsemiring

theorem subring.to_subsemiring_strict_mono {R : Type u} [ring R] :

monotone subring.to_subsemiring

theorem subring.to_subsemiring_mono {R : Type u} [ring R] :

function.injective subring.to_add_subgroup

theorem subring.to_add_subgroup_injective {R : Type u} [ring R] :

strict_mono subring.to_add_subgroup

theorem subring.to_add_subgroup_strict_mono {R : Type u} [ring R] :

monotone subring.to_add_subgroup

theorem subring.to_add_subgroup_mono {R : Type u} [ring R] :

function.injective subring.to_submonoid

theorem subring.to_submonoid_injective {R : Type u} [ring R] :

strict_mono subring.to_submonoid

theorem subring.to_submonoid_strict_mono {R : Type u} [ring R] :

monotone subring.to_submonoid

theorem subring.to_submonoid_mono {R : Type u} [ring R] :

@[protected]

def subring.mk' {R : Type u} [ring R] (s : set R) (sm : submonoid R) (sa : add_subgroup R) (hm : ↑sm = s) (ha : ↑sa = s) :

Construct a subring R from a set s, a submonoid sm, and an additive subgroup sa such that x ∈ s ↔ x ∈ sm ↔ x ∈ sa.

Equations

subring.mk' s sm sa hm ha = {carrier := s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[simp]

theorem subring.coe_mk' {R : Type u} [ring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_subgroup R} (ha : ↑sa = s) :

↑(subring.mk' s sm sa hm ha) = s

@[simp]

theorem subring.mem_mk' {R : Type u} [ring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_subgroup R} (ha : ↑sa = s) {x : R} :

x ∈ subring.mk' s sm sa hm ha ↔ x ∈ s

@[simp]

theorem subring.mk'_to_submonoid {R : Type u} [ring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_subgroup R} (ha : ↑sa = s) :

(subring.mk' s sm sa hm ha).to_submonoid = sm

@[simp]

theorem subring.mk'_to_add_subgroup {R : Type u} [ring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_subgroup R} (ha : ↑sa = s) :

(subring.mk' s sm sa hm ha).to_add_subgroup = sa

def subsemiring.to_subring {R : Type u} [ring R] (s : subsemiring R) (hneg : -1 ∈ s) :

A subsemiring containing -1 is a subring.

Equations

s.to_subring hneg = {carrier := s.to_submonoid.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[protected]

theorem subring.one_mem {R : Type u} [ring R] (s : subring R) :

1 ∈ s

A subring contains the ring's 1.

@[protected]

theorem subring.zero_mem {R : Type u} [ring R] (s : subring R) :

0 ∈ s

A subring contains the ring's 0.

@[protected]

theorem subring.mul_mem {R : Type u} [ring R] (s : subring R) {x y : R} :

x ∈ s → y ∈ s → x * y ∈ s

A subring is closed under multiplication.

@[protected]

theorem subring.add_mem {R : Type u} [ring R] (s : subring R) {x y : R} :

x ∈ s → y ∈ s → x + y ∈ s

A subring is closed under addition.

@[protected]

theorem subring.neg_mem {R : Type u} [ring R] (s : subring R) {x : R} :

x ∈ s → -x ∈ s

A subring is closed under negation.

@[protected]

theorem subring.sub_mem {R : Type u} [ring R] (s : subring R) {x y : R} (hx : x ∈ s) (hy : y ∈ s) :

x - y ∈ s

A subring is closed under subtraction

@[protected]

theorem subring.list_prod_mem {R : Type u} [ring R] (s : subring R) {l : list R} :

(∀ (x : R), x ∈ l → x ∈ s) → l.prod ∈ s

Product of a list of elements in a subring is in the subring.

@[protected]

theorem subring.list_sum_mem {R : Type u} [ring R] (s : subring R) {l : list R} :

(∀ (x : R), x ∈ l → x ∈ s) → l.sum ∈ s

Sum of a list of elements in a subring is in the subring.

@[protected]

theorem subring.multiset_prod_mem {R : Type u_1} [comm_ring R] (s : subring R) (m : multiset R) :

(∀ (a : R), a ∈ m → a ∈ s) → m.prod ∈ s

Product of a multiset of elements in a subring of a comm_ring is in the subring.

@[protected]

theorem subring.multiset_sum_mem {R : Type u_1} [ring R] (s : subring R) (m : multiset R) :

(∀ (a : R), a ∈ m → a ∈ s) → m.sum ∈ s

Sum of a multiset of elements in an subring of a ring is in the subring.

@[protected]

theorem subring.prod_mem {R : Type u_1} [comm_ring R] (s : subring R) {ι : Type u_2} {t : finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

t.prod (λ (i : ι), f i) ∈ s

Product of elements of a subring of a comm_ring indexed by a finset is in the subring.

@[protected]

theorem subring.sum_mem {R : Type u_1} [ring R] (s : subring R) {ι : Type u_2} {t : finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

t.sum (λ (i : ι), f i) ∈ s

Sum of elements in a subring of a ring indexed by a finset is in the subring.

@[protected, instance]

def subring.to_ring {R : Type u} [ring R] (s : subring R) :

ring ↥s

A subring of a ring inherits a ring structure

Equations

s.to_ring = function.injective.ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected]

theorem subring.zsmul_mem {R : Type u} [ring R] (s : subring R) {x : R} (hx : x ∈ s) (n : ℤ) :

n • x ∈ s

@[protected]

theorem subring.pow_mem {R : Type u} [ring R] (s : subring R) {x : R} (hx : x ∈ s) (n : ℕ) :

x ^ n ∈ s

@[simp, norm_cast]

theorem subring.coe_add {R : Type u} [ring R] (s : subring R) (x y : ↥s) :

↑(x + y) = ↑x + ↑y

@[simp, norm_cast]

theorem subring.coe_neg {R : Type u} [ring R] (s : subring R) (x : ↥s) :

↑-x = -↑x

@[simp, norm_cast]

theorem subring.coe_mul {R : Type u} [ring R] (s : subring R) (x y : ↥s) :

↑(x * y) = ↑x * ↑y

@[simp, norm_cast]

theorem subring.coe_zero {R : Type u} [ring R] (s : subring R) :

↑0 = 0

@[simp, norm_cast]

theorem subring.coe_one {R : Type u} [ring R] (s : subring R) :

↑1 = 1

@[simp, norm_cast]

theorem subring.coe_pow {R : Type u} [ring R] (s : subring R) (x : ↥s) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

@[simp]

theorem subring.coe_eq_zero_iff {R : Type u} [ring R] (s : subring R) {x : ↥s} :

↑x = 0 ↔ x = 0

@[protected, instance]

def subring.to_comm_ring {R : Type u_1} [comm_ring R] (s : subring R) :

comm_ring ↥s

A subring of a comm_ring is a comm_ring.

Equations

s.to_comm_ring = function.injective.comm_ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring.nontrivial {R : Type u_1} [ring R] [nontrivial R] (s : subring R) :

nontrivial ↥s

A subring of a non-trivial ring is non-trivial.

@[protected, instance]

def subring.no_zero_divisors {R : Type u_1} [ring R] [no_zero_divisors R] (s : subring R) :

no_zero_divisors ↥s

A subring of a ring with no zero divisors has no zero divisors.

@[protected, instance]

def subring.is_domain {R : Type u_1} [ring R] [is_domain R] (s : subring R) :

is_domain ↥s

A subring of a domain is a domain.

@[protected, instance]

def subring.to_ordered_ring {R : Type u_1} [ordered_ring R] (s : subring R) :

ordered_ring ↥s

A subring of an ordered_ring is an ordered_ring.

Equations

s.to_ordered_ring = function.injective.ordered_ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring.to_ordered_comm_ring {R : Type u_1} [ordered_comm_ring R] (s : subring R) :

ordered_comm_ring ↥s

A subring of an ordered_comm_ring is an ordered_comm_ring.

Equations

s.to_ordered_comm_ring = function.injective.ordered_comm_ring coe _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subring.to_linear_ordered_ring {R : Type u_1} [linear_ordered_ring R] (s : subring R) :

linear_ordered_ring ↥s

A subring of a linear_ordered_ring is a linear_ordered_ring.

Equations

s.to_linear_ordered_ring = function.injective.linear_ordered_ring coe _ _ _ _ _ _ _ _ _ _ _ _ _ _

linear_ordered_comm_ring ↥s

@[protected, instance]

def subring.to_linear_ordered_comm_ring {R : Type u_1} [linear_ordered_comm_ring R] (s : subring R) :

A subring of a linear_ordered_comm_ring is a linear_ordered_comm_ring.

Equations

s.to_linear_ordered_comm_ring = function.injective.linear_ordered_comm_ring coe _ _ _ _ _ _ _ _ _ _ _ _ _ _

def subring.subtype {R : Type u} [ring R] (s : subring R) :

↥s →+* R

The natural ring hom from a subring of ring R to R.

Equations

s.subtype = {to_fun := coe coe_to_lift, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

@[simp]

theorem subring.coe_subtype {R : Type u} [ring R] (s : subring R) :

⇑(s.subtype) = coe

@[simp, norm_cast]

theorem subring.coe_nat_cast {R : Type u} [ring R] (s : subring R) (n : ℕ) :

@[simp, norm_cast]

theorem subring.coe_int_cast {R : Type u} [ring R] (s : subring R) (n : ℤ) :

@[simp]

theorem subring.mem_to_submonoid {R : Type u} [ring R] {s : subring R} {x : R} :

x ∈ s.to_submonoid ↔ x ∈ s

@[simp]

theorem subring.coe_to_submonoid {R : Type u} [ring R] (s : subring R) :

↑(s.to_submonoid) = ↑s

@[simp]

theorem subring.mem_to_add_subgroup {R : Type u} [ring R] {s : subring R} {x : R} :

x ∈ s.to_add_subgroup ↔ x ∈ s

@[simp]

theorem subring.coe_to_add_subgroup {R : Type u} [ring R] (s : subring R) :

↑(s.to_add_subgroup) = ↑s

@[protected, instance]

def subring.has_top {R : Type u} [ring R] :

has_top (subring R)

The subring R of the ring R.

Equations

subring.has_top = {top := {carrier := ⊤.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}}

@[simp]

theorem subring.mem_top {R : Type u} [ring R] (x : R) :

x ∈ ⊤

@[simp]

theorem subring.coe_top {R : Type u} [ring R] :

↑⊤ = set.univ

def subring.top_equiv {R : Type u} [ring R] :

↥⊤ ≃+* R

The ring equiv between the top element of subring R and R.

Equations

subring.top_equiv = subsemiring.top_equiv

@[simp]

theorem subring.top_equiv_apply {R : Type u} [ring R] (r : ↥⊤) :

⇑subring.top_equiv r = ↑r

@[simp]

theorem subring.top_equiv_symm_apply_coe {R : Type u} [ring R] (r : R) :

↑(⇑(subring.top_equiv.symm) r) = r

def subring.comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : subring S) :

The preimage of a subring along a ring homomorphism is a subring.

Equations

subring.comap f s = {carrier := ⇑f ⁻¹' s.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[simp]

theorem subring.coe_comap {R : Type u} {S : Type v} [ring R] [ring S] (s : subring S) (f : R →+* S) :

↑(subring.comap f s) = ⇑f ⁻¹' ↑s

@[simp]

theorem subring.mem_comap {R : Type u} {S : Type v} [ring R] [ring S] {s : subring S} {f : R →+* S} {x : R} :

x ∈ subring.comap f s ↔ ⇑f x ∈ s

theorem subring.comap_comap {R : Type u} {S : Type v} {T : Type w} [ring R] [ring S] [ring T] (s : subring T) (g : S →+* T) (f : R →+* S) :

subring.comap f (subring.comap g s) = subring.comap (g.comp f) s

def subring.map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : subring R) :

subring S

The image of a subring along a ring homomorphism is a subring.

Equations

subring.map f s = {carrier := ⇑f '' s.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[simp]

theorem subring.coe_map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : subring R) :

↑(subring.map f s) = ⇑f '' ↑s

@[simp]

theorem subring.mem_map {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {s : subring R} {y : S} :

y ∈ subring.map f s ↔ ∃ (x : R) (H : x ∈ s), ⇑f x = y

@[simp]

theorem subring.map_id {R : Type u} [ring R] (s : subring R) :

subring.map (ring_hom.id R) s = s

theorem subring.map_map {R : Type u} {S : Type v} {T : Type w} [ring R] [ring S] [ring T] (s : subring R) (g : S →+* T) (f : R →+* S) :

subring.map g (subring.map f s) = subring.map (g.comp f) s

theorem subring.map_le_iff_le_comap {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {s : subring R} {t : subring S} :

subring.map f s ≤ t ↔ s ≤ subring.comap f t

theorem subring.gc_map_comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

galois_connection (subring.map f) (subring.comap f)

noncomputable def subring.equiv_map_of_injective {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) (f : R →+* S) (hf : function.injective ⇑f) :

↥s ≃+* ↥(subring.map f s)

A subring is isomorphic to its image under an injective function

Equations

s.equiv_map_of_injective f hf = {to_fun := (equiv.set.image ⇑f ↑s hf).to_fun, inv_fun := (equiv.set.image ⇑f ↑s hf).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

@[simp]

theorem subring.coe_equiv_map_of_injective_apply {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) (f : R →+* S) (hf : function.injective ⇑f) (x : ↥s) :

↑(⇑(s.equiv_map_of_injective f hf) x) = ⇑f ↑x

def ring_hom.range {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

subring S

The range of a ring homomorphism, as a subring of the target. See Note [range copy pattern].

Equations

f.range = (subring.map f ⊤).copy (set.range ⇑f) _

Instances for ↥ring_hom.range

@[simp]

theorem ring_hom.coe_range {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

↑(f.range) = set.range ⇑f

@[simp]

theorem ring_hom.mem_range {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {y : S} :

y ∈ f.range ↔ ∃ (x : R), ⇑f x = y

theorem ring_hom.range_eq_map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

f.range = subring.map f ⊤

theorem ring_hom.mem_range_self {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (x : R) :

⇑f x ∈ f.range

theorem ring_hom.map_range {R : Type u} {S : Type v} {T : Type w} [ring R] [ring S] [ring T] (g : S →+* T) (f : R →+* S) :

subring.map g f.range = (g.comp f).range

@[protected, instance]

def ring_hom.fintype_range {R : Type u} {S : Type v} [ring R] [ring S] [fintype R] [decidable_eq S] (f : R →+* S) :

fintype ↥(f.range)

The range of a ring homomorphism is a fintype, if the domain is a fintype. Note: this instance can form a diamond with subtype.fintype in the presence of fintype S.

Equations

f.fintype_range = set.fintype_range ⇑f

@[protected, instance]

def subring.has_bot {R : Type u} [ring R] :

has_bot (subring R)

Equations

subring.has_bot = {bot := (int.cast_ring_hom R).range}

@[protected, instance]

def subring.inhabited {R : Type u} [ring R] :

inhabited (subring R)

Equations

subring.inhabited = {default := ⊥}

theorem subring.coe_bot {R : Type u} [ring R] :

↑⊥ = set.range coe

theorem subring.mem_bot {R : Type u} [ring R] {x : R} :

x ∈ ⊥ ↔ ∃ (n : ℤ), ↑n = x

@[protected, instance]

def subring.has_inf {R : Type u} [ring R] :

has_inf (subring R)

The inf of two subrings is their intersection.

Equations

subring.has_inf = {inf := λ (s t : subring R), {carrier := ↑s ∩ ↑t, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}}

@[simp]

theorem subring.coe_inf {R : Type u} [ring R] (p p' : subring R) :

↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem subring.mem_inf {R : Type u} [ring R] {p p' : subring R} {x : R} :

x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

@[protected, instance]

def subring.has_Inf {R : Type u} [ring R] :

has_Inf (subring R)

Equations

subring.has_Inf = {Inf := λ (s : set (subring R)), subring.mk' (⋂ (t : subring R) (H : t ∈ s), ↑t) (⨅ (t : subring R) (H : t ∈ s), t.to_submonoid) (⨅ (t : subring R) (H : t ∈ s), t.to_add_subgroup) _ _}

@[simp, norm_cast]

theorem subring.coe_Inf {R : Type u} [ring R] (S : set (subring R)) :

↑(has_Inf.Inf S) = ⋂ (s : subring R) (H : s ∈ S), ↑s

theorem subring.mem_Inf {R : Type u} [ring R] {S : set (subring R)} {x : R} :

x ∈ has_Inf.Inf S ↔ ∀ (p : subring R), p ∈ S → x ∈ p

@[simp, norm_cast]

theorem subring.coe_infi {R : Type u} [ring R] {ι : Sort u_1} {S : ι → subring R} :

(↑⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

theorem subring.mem_infi {R : Type u} [ring R] {ι : Sort u_1} {S : ι → subring R} {x : R} :

(x ∈ ⨅ (i : ι), S i) ↔ ∀ (i : ι), x ∈ S i

@[simp]

theorem subring.Inf_to_submonoid {R : Type u} [ring R] (s : set (subring R)) :

(has_Inf.Inf s).to_submonoid = ⨅ (t : subring R) (H : t ∈ s), t.to_submonoid

@[simp]

theorem subring.Inf_to_add_subgroup {R : Type u} [ring R] (s : set (subring R)) :

(has_Inf.Inf s).to_add_subgroup = ⨅ (t : subring R) (H : t ∈ s), t.to_add_subgroup

@[protected, instance]

def subring.complete_lattice {R : Type u} [ring R] :

complete_lattice (subring R)

Subrings of a ring form a complete lattice.

Equations

subring.complete_lattice = {sup := complete_lattice.sup (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), le := complete_lattice.le (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), lt := complete_lattice.lt (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := has_inf.inf subring.has_inf, inf_le_left := _, inf_le_right := _, le_inf := _, Sup := complete_lattice.Sup (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), Inf_le := _, le_Inf := _, top := ⊤, bot := ⊥, le_top := _, bot_le := _}

theorem subring.eq_top_iff' {R : Type u} [ring R] (A : subring R) :

A = ⊤ ↔ ∀ (x : R), x ∈ A

def subring.center (R : Type u) [ring R] :

The center of a ring R is the set of elements that commute with everything in R

Equations

subring.center R = {carrier := set.center R (distrib.to_has_mul R), mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

Instances for ↥subring.center

theorem subring.coe_center (R : Type u) [ring R] :

↑(subring.center R) = set.center R

@[simp]

theorem subring.center_to_subsemiring (R : Type u) [ring R] :

(subring.center R).to_subsemiring = subsemiring.center R

theorem subring.mem_center_iff {R : Type u} [ring R] {z : R} :

z ∈ subring.center R ↔ ∀ (g : R), g * z = z * g

@[protected, instance]

def subring.decidable_mem_center {R : Type u} [ring R] [decidable_eq R] [fintype R] :

decidable_pred (λ (_x : R), _x ∈ subring.center R)

Equations

subring.decidable_mem_center = λ (_x : R), decidable_of_iff' (∀ (g : R), g * _x = _x * g) subring.mem_center_iff

@[simp]

theorem subring.center_eq_top (R : Type u_1) [comm_ring R] :

subring.center R = ⊤

@[protected, instance]

def subring.center.comm_ring {R : Type u} [ring R] :

comm_ring ↥(subring.center R)

The center is commutative.

Equations

subring.center.comm_ring = {add := comm_semiring.add subsemiring.center.comm_semiring, add_assoc := _, zero := comm_semiring.zero subsemiring.center.comm_semiring, zero_add := _, add_zero := _, nsmul := comm_semiring.nsmul subsemiring.center.comm_semiring, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg (subring.center R).to_ring, sub := ring.sub (subring.center R).to_ring, sub_eq_add_neg := _, zsmul := ring.zsmul (subring.center R).to_ring, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _, int_cast := ring.int_cast (subring.center R).to_ring, nat_cast := comm_semiring.nat_cast subsemiring.center.comm_semiring, one := comm_semiring.one subsemiring.center.comm_semiring, nat_cast_zero := _, nat_cast_succ := _, int_cast_of_nat := _, int_cast_neg_succ_of_nat := _, mul := comm_semiring.mul subsemiring.center.comm_semiring, mul_assoc := _, one_mul := _, mul_one := _, npow := comm_semiring.npow subsemiring.center.comm_semiring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _}

@[protected, instance]

def subring.center.field {K : Type u} [division_ring K] :

field ↥(subring.center K)

Equations

subring.center.field = {add := comm_ring.add subring.center.comm_ring, add_assoc := _, zero := comm_ring.zero subring.center.comm_ring, zero_add := _, add_zero := _, nsmul := comm_ring.nsmul subring.center.comm_ring, nsmul_zero' := _, nsmul_succ' := _, neg := comm_ring.neg subring.center.comm_ring, sub := comm_ring.sub subring.center.comm_ring, sub_eq_add_neg := _, zsmul := comm_ring.zsmul subring.center.comm_ring, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _, int_cast := comm_ring.int_cast subring.center.comm_ring, nat_cast := comm_ring.nat_cast subring.center.comm_ring, one := comm_ring.one subring.center.comm_ring, nat_cast_zero := _, nat_cast_succ := _, int_cast_of_nat := _, int_cast_neg_succ_of_nat := _, mul := comm_ring.mul subring.center.comm_ring, mul_assoc := _, one_mul := _, mul_one := _, npow := comm_ring.npow subring.center.comm_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _, inv := λ (a : ↥(subring.center K)), ⟨(↑a)⁻¹, _⟩, div := λ (a b : ↥(subring.center K)), ⟨↑a / ↑b, _⟩, div_eq_mul_inv := _, zpow := division_ring.zpow._default comm_ring.mul subring.center.field._proof_27 comm_ring.one subring.center.field._proof_28 subring.center.field._proof_29 comm_ring.npow subring.center.field._proof_30 subring.center.field._proof_31 (λ (a : ↥(subring.center K)), ⟨(↑a)⁻¹, _⟩), zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, exists_pair_ne := _, rat_cast := division_ring.rat_cast._default comm_ring.add subring.center.field._proof_36 comm_ring.zero subring.center.field._proof_37 subring.center.field._proof_38 comm_ring.nsmul subring.center.field._proof_39 subring.center.field._proof_40 comm_ring.neg comm_ring.sub subring.center.field._proof_41 comm_ring.zsmul subring.center.field._proof_42 subring.center.field._proof_43 subring.center.field._proof_44 subring.center.field._proof_45 subring.center.field._proof_46 comm_ring.int_cast comm_ring.nat_cast comm_ring.one subring.center.field._proof_47 subring.center.field._proof_48 subring.center.field._proof_49 subring.center.field._proof_50 comm_ring.mul subring.center.field._proof_51 subring.center.field._proof_52 subring.center.field._proof_53 comm_ring.npow subring.center.field._proof_54 subring.center.field._proof_55 subring.center.field._proof_56 subring.center.field._proof_57 (λ (a : ↥(subring.center K)), ⟨(↑a)⁻¹, _⟩) (λ (a b : ↥(subring.center K)), ⟨↑a / ↑b, _⟩) subring.center.field._proof_58 (division_ring.zpow._default comm_ring.mul subring.center.field._proof_27 comm_ring.one subring.center.field._proof_28 subring.center.field._proof_29 comm_ring.npow subring.center.field._proof_30 subring.center.field._proof_31 (λ (a : ↥(subring.center K)), ⟨(↑a)⁻¹, _⟩)) subring.center.field._proof_59 subring.center.field._proof_60 subring.center.field._proof_61, mul_inv_cancel := _, inv_zero := _, rat_cast_mk := _, qsmul := division_ring.qsmul._default (… subring.center.field._proof_42 subring.center.field._proof_43 subring.center.field._proof_44 subring.center.field._proof_45 subring.center.field._proof_46 comm_ring.int_cast comm_ring.nat_cast comm_ring.one subring.center.field._proof_47 subring.center.field._proof_48 subring.center.field._proof_49 subring.center.field._proof_50 comm_ring.mul subring.center.field._proof_51 subring.center.field._proof_52 subring.center.field._proof_53 comm_ring.npow subring.center.field._proof_54 subring.center.field._proof_55 subring.center.field._proof_56 subring.center.field._proof_57 (λ (a : ↥(subring.center K)), ⟨(↑a)⁻¹, _⟩) (λ (a b : ↥(subring.center K)), ⟨↑a / ↑b, _⟩) subring.center.field._proof_58 (division_ring.zpow._default comm_ring.mul subring.center.field._proof_27 comm_ring.one subring.center.field._proof_28 subring.center.field._proof_29 comm_ring.npow subring.center.field._proof_30 subring.center.field._proof_31 (λ (a : ↥(subring.center K)), ⟨(↑a)⁻¹, _⟩)) subring.center.field._proof_59 subring.center.field._proof_60 subring.center.field._proof_61) comm_ring.add subring.center.field._proof_65 comm_ring.zero subring.center.field._proof_66 subring.center.field._proof_67 comm_ring.nsmul subring.center.field._proof_68 subring.center.field._proof_69 comm_ring.neg comm_ring.sub subring.center.field._proof_70 comm_ring.zsmul subring.center.field._proof_71 subring.center.field._proof_72 subring.center.field._proof_73 subring.center.field._proof_74 subring.center.field._proof_75 comm_ring.int_cast comm_ring.nat_cast comm_ring.one subring.center.field._proof_76 subring.center.field._proof_77 subring.center.field._proof_78 subring.center.field._proof_79 comm_ring.mul subring.center.field._proof_80 subring.center.field._proof_81 subring.center.field._proof_82 comm_ring.npow subring.center.field._proof_83 subring.center.field._proof_84 subring.center.field._proof_85 subring.center.field._proof_86, qsmul_eq_mul' := _}

@[simp]

theorem subring.center.coe_inv {K : Type u} [division_ring K] (a : ↥(subring.center K)) :

↑a⁻¹ = (↑a)⁻¹

@[simp]

theorem subring.center.coe_div {K : Type u} [division_ring K] (a b : ↥(subring.center K)) :

↑(a / b) = ↑a / ↑b

def subring.centralizer {R : Type u} [ring R] (s : set R) :

The centralizer of a set inside a ring as a subring.

Equations

subring.centralizer s = {carrier := (subsemiring.centralizer s).carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[simp, norm_cast]

theorem subring.coe_centralizer {R : Type u} [ring R] (s : set R) :

↑(subring.centralizer s) = s.centralizer

theorem subring.centralizer_to_submonoid {R : Type u} [ring R] (s : set R) :

(subring.centralizer s).to_submonoid = submonoid.centralizer s

theorem subring.centralizer_to_subsemiring {R : Type u} [ring R] (s : set R) :

(subring.centralizer s).to_subsemiring = subsemiring.centralizer s

theorem subring.mem_centralizer_iff {R : Type u} [ring R] {s : set R} {z : R} :

z ∈ subring.centralizer s ↔ ∀ (g : R), g ∈ s → g * z = z * g

theorem subring.center_le_centralizer {R : Type u} [ring R] (s : set R) :

subring.center R ≤ subring.centralizer s

theorem subring.centralizer_le {R : Type u} [ring R] (s t : set R) (h : s ⊆ t) :

subring.centralizer t ≤ subring.centralizer s

@[simp]

theorem subring.centralizer_eq_top_iff_subset {R : Type u} [ring R] {s : set R} :

subring.centralizer s = ⊤ ↔ s ⊆ ↑(subring.center R)

@[simp]

theorem subring.centralizer_univ {R : Type u} [ring R] :

subring.centralizer set.univ = subring.center R

def subring.closure {R : Type u} [ring R] (s : set R) :

The subring generated by a set.

Equations

subring.closure s = has_Inf.Inf {S : subring R | s ⊆ ↑S}

theorem subring.mem_closure {R : Type u} [ring R] {x : R} {s : set R} :

x ∈ subring.closure s ↔ ∀ (S : subring R), s ⊆ ↑S → x ∈ S

@[simp]

theorem subring.subset_closure {R : Type u} [ring R] {s : set R} :

s ⊆ ↑(subring.closure s)

The subring generated by a set includes the set.

theorem subring.not_mem_of_not_mem_closure {R : Type u} [ring R] {s : set R} {P : R} (hP : P ∉ subring.closure s) :

P ∉ s

@[simp]

theorem subring.closure_le {R : Type u} [ring R] {s : set R} {t : subring R} :

subring.closure s ≤ t ↔ s ⊆ ↑t

A subring t includes closure s if and only if it includes s.

theorem subring.closure_mono {R : Type u} [ring R] ⦃s t : set R⦄ (h : s ⊆ t) :

subring.closure s ≤ subring.closure t

Subring closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

theorem subring.closure_eq_of_le {R : Type u} [ring R] {s : set R} {t : subring R} (h₁ : s ⊆ ↑t) (h₂ : t ≤ subring.closure s) :

subring.closure s = t

theorem subring.closure_induction {R : Type u} [ring R] {s : set R} {p : R → Prop} {x : R} (h : x ∈ subring.closure s) (Hs : ∀ (x : R), x ∈ s → p x) (H0 : p 0) (H1 : p 1) (Hadd : ∀ (x y : R), p x → p y → p (x + y)) (Hneg : ∀ (x : R), p x → p (-x)) (Hmul : ∀ (x y : R), p x → p y → p (x * y)) :

p x

An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition, negation, and multiplication, then p holds for all elements of the closure of s.

theorem subring.closure_induction₂ {R : Type u} [ring R] {s : set R} {p : R → R → Prop} {a b : R} (ha : a ∈ subring.closure s) (hb : b ∈ subring.closure s) (Hs : ∀ (x : R), x ∈ s → ∀ (y : R), y ∈ s → p x y) (H0_left : ∀ (x : R), p 0 x) (H0_right : ∀ (x : R), p x 0) (H1_left : ∀ (x : R), p 1 x) (H1_right : ∀ (x : R), p x 1) (Hneg_left : ∀ (x y : R), p x y → p (-x) y) (Hneg_right : ∀ (x y : R), p x y → p x (-y)) (Hadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y) (Hadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)) (Hmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y) (Hmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)) :

p a b

An induction principle for closure membership, for predicates with two arguments.

theorem subring.mem_closure_iff {R : Type u} [ring R] {s : set R} {x : R} :

x ∈ subring.closure s ↔ x ∈ add_subgroup.closure ↑(submonoid.closure s)

def subring.closure_comm_ring_of_comm {R : Type u} [ring R] {s : set R} (hcomm : ∀ (a : R), a ∈ s → ∀ (b : R), b ∈ s → a * b = b * a) :

comm_ring ↥(subring.closure s)

If all elements of s : set A commute pairwise, then closure s is a commutative ring.

Equations

subring.closure_comm_ring_of_comm hcomm = {add := ring.add (subring.closure s).to_ring, add_assoc := _, zero := ring.zero (subring.closure s).to_ring, zero_add := _, add_zero := _, nsmul := ring.nsmul (subring.closure s).to_ring, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg (subring.closure s).to_ring, sub := ring.sub (subring.closure s).to_ring, sub_eq_add_neg := _, zsmul := ring.zsmul (subring.closure s).to_ring, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _, int_cast := ring.int_cast (subring.closure s).to_ring, nat_cast := ring.nat_cast (subring.closure s).to_ring, one := ring.one (subring.closure s).to_ring, nat_cast_zero := _, nat_cast_succ := _, int_cast_of_nat := _, int_cast_neg_succ_of_nat := _, mul := ring.mul (subring.closure s).to_ring, mul_assoc := _, one_mul := _, mul_one := _, npow := ring.npow (subring.closure s).to_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _}

theorem subring.exists_list_of_mem_closure {R : Type u} [ring R] {s : set R} {x : R} (h : x ∈ subring.closure s) :

∃ (L : list (list R)), (∀ (t : list R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ (list.map list.prod L).sum = x

galois_insertion subring.closure coe

@[protected]

def subring.gi (R : Type u) [ring R] :

closure forms a Galois insertion with the coercion to set.

Equations

subring.gi R = {choice := λ (s : set R) (_x : ↑(subring.closure s) ≤ s), subring.closure s, gc := _, le_l_u := _, choice_eq := _}

theorem subring.closure_eq {R : Type u} [ring R] (s : subring R) :

subring.closure ↑s = s

Closure of a subring S equals S.

@[simp]

theorem subring.closure_empty {R : Type u} [ring R] :

subring.closure ∅ = ⊥

subring.closure set.univ = ⊤

@[simp]

theorem subring.closure_univ {R : Type u} [ring R] :

theorem subring.closure_union {R : Type u} [ring R] (s t : set R) :

subring.closure (s ∪ t) = subring.closure s ⊔ subring.closure t

theorem subring.closure_Union {R : Type u} [ring R] {ι : Sort u_1} (s : ι → set R) :

subring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), subring.closure (s i)

theorem subring.closure_sUnion {R : Type u} [ring R] (s : set (set R)) :

subring.closure (⋃₀ s) = ⨆ (t : set R) (H : t ∈ s), subring.closure t

theorem subring.map_sup {R : Type u} {S : Type v} [ring R] [ring S] (s t : subring R) (f : R →+* S) :

subring.map f (s ⊔ t) = subring.map f s ⊔ subring.map f t

theorem subring.map_supr {R : Type u} {S : Type v} [ring R] [ring S] {ι : Sort u_1} (f : R →+* S) (s : ι → subring R) :

subring.map f (supr s) = ⨆ (i : ι), subring.map f (s i)

theorem subring.comap_inf {R : Type u} {S : Type v} [ring R] [ring S] (s t : subring S) (f : R →+* S) :

subring.comap f (s ⊓ t) = subring.comap f s ⊓ subring.comap f t

theorem subring.comap_infi {R : Type u} {S : Type v} [ring R] [ring S] {ι : Sort u_1} (f : R →+* S) (s : ι → subring S) :

subring.comap f (infi s) = ⨅ (i : ι), subring.comap f (s i)

@[simp]

theorem subring.map_bot {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

subring.map f ⊥ = ⊥

@[simp]

theorem subring.comap_top {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

subring.comap f ⊤ = ⊤

def subring.prod {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) (t : subring S) :

subring (R × S)

Given subrings s, t of rings R, S respectively, s.prod t is s ×̂ t as a subring of R × S.

Equations

s.prod t = {carrier := ↑s ×ˢ ↑t, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[norm_cast]

theorem subring.coe_prod {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) (t : subring S) :

↑(s.prod t) = ↑s ×ˢ ↑t

theorem subring.mem_prod {R : Type u} {S : Type v} [ring R] [ring S] {s : subring R} {t : subring S} {p : R × S} :

p ∈ s.prod t ↔ p.fst ∈ s ∧ p.snd ∈ t

theorem subring.prod_mono {R : Type u} {S : Type v} [ring R] [ring S] ⦃s₁ s₂ : subring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : subring S⦄ (ht : t₁ ≤ t₂) :

s₁.prod t₁ ≤ s₂.prod t₂

theorem subring.prod_mono_right {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) :

monotone (λ (t : subring S), s.prod t)

theorem subring.prod_mono_left {R : Type u} {S : Type v} [ring R] [ring S] (t : subring S) :

monotone (λ (s : subring R), s.prod t)

theorem subring.prod_top {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) :

s.prod ⊤ = subring.comap (ring_hom.fst R S) s

theorem subring.top_prod {R : Type u} {S : Type v} [ring R] [ring S] (s : subring S) :

⊤.prod s = subring.comap (ring_hom.snd R S) s

@[simp]

theorem subring.top_prod_top {R : Type u} {S : Type v} [ring R] [ring S] :

⊤.prod ⊤ = ⊤

def subring.prod_equiv {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) (t : subring S) :

↥(s.prod t) ≃+* ↥s × ↥t

Product of subrings is isomorphic to their product as rings.

Equations

s.prod_equiv t = {to_fun := (equiv.set.prod ↑s ↑t).to_fun, inv_fun := (equiv.set.prod ↑s ↑t).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

theorem subring.mem_supr_of_directed {R : Type u} [ring R] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subring R} (hS : directed has_le.le S) {x : R} :

(x ∈ ⨆ (i : ι), S i) ↔ ∃ (i : ι), x ∈ S i

The underlying set of a non-empty directed Sup of subrings is just a union of the subrings. Note that this fails without the directedness assumption (the union of two subrings is typically not a subring)

theorem subring.coe_supr_of_directed {R : Type u} [ring R] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subring R} (hS : directed has_le.le S) :

(↑⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem subring.mem_Sup_of_directed_on {R : Type u} [ring R] {S : set (subring R)} (Sne : S.nonempty) (hS : directed_on has_le.le S) {x : R} :

x ∈ has_Sup.Sup S ↔ ∃ (s : subring R) (H : s ∈ S), x ∈ s

theorem subring.coe_Sup_of_directed_on {R : Type u} [ring R] {S : set (subring R)} (Sne : S.nonempty) (hS : directed_on has_le.le S) :

↑(has_Sup.Sup S) = ⋃ (s : subring R) (H : s ∈ S), ↑s

theorem subring.mem_map_equiv {R : Type u} {S : Type v} [ring R] [ring S] {f : R ≃+* S} {K : subring R} {x : S} :

x ∈ subring.map ↑f K ↔ ⇑(f.symm) x ∈ K

theorem subring.map_equiv_eq_comap_symm {R : Type u} {S : Type v} [ring R] [ring S] (f : R ≃+* S) (K : subring R) :

subring.map ↑f K = subring.comap ↑(f.symm) K

theorem subring.comap_equiv_eq_map_symm {R : Type u} {S : Type v} [ring R] [ring S] (f : R ≃+* S) (K : subring S) :

subring.comap ↑f K = subring.map ↑(f.symm) K

def ring_hom.range_restrict {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

R →+* ↥(f.range)

Restriction of a ring homomorphism to its range interpreted as a subsemiring.

This is the bundled version of set.range_factorization.

Equations

f.range_restrict = f.cod_restrict f.range _

@[simp]

theorem ring_hom.coe_range_restrict {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (x : R) :

↑(⇑(f.range_restrict) x) = ⇑f x

theorem ring_hom.range_restrict_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

function.surjective ⇑(f.range_restrict)

theorem ring_hom.range_top_iff_surjective {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} :

f.range = ⊤ ↔ function.surjective ⇑f

theorem ring_hom.range_top_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) :

f.range = ⊤

The range of a surjective ring homomorphism is the whole of the codomain.

def ring_hom.eq_locus {R : Type u} {S : Type v} [ring R] [ring S] (f g : R →+* S) :

The subring of elements x : R such that f x = g x, i.e., the equalizer of f and g as a subring of R

Equations

f.eq_locus g = {carrier := {x : R | ⇑f x = ⇑g x}, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

@[simp]

theorem ring_hom.eq_locus_same {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

f.eq_locus f = ⊤

theorem ring_hom.eq_on_set_closure {R : Type u} {S : Type v} [ring R] [ring S] {f g : R →+* S} {s : set R} (h : set.eq_on ⇑f ⇑g s) :

set.eq_on ⇑f ⇑g ↑(subring.closure s)

If two ring homomorphisms are equal on a set, then they are equal on its subring closure.

theorem ring_hom.eq_of_eq_on_set_top {R : Type u} {S : Type v} [ring R] [ring S] {f g : R →+* S} (h : set.eq_on ⇑f ⇑g ↑⊤) :

f = g

theorem ring_hom.eq_of_eq_on_set_dense {R : Type u} {S : Type v} [ring R] [ring S] {s : set R} (hs : subring.closure s = ⊤) {f g : R →+* S} (h : set.eq_on ⇑f ⇑g s) :

f = g

theorem ring_hom.closure_preimage_le {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set S) :

subring.closure (⇑f ⁻¹' s) ≤ subring.comap f (subring.closure s)

theorem ring_hom.map_closure {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set R) :

subring.map f (subring.closure s) = subring.closure (⇑f '' s)

The image under a ring homomorphism of the subring generated by a set equals the subring generated by the image of the set.

def subring.inclusion {R : Type u} [ring R] {S T : subring R} (h : S ≤ T) :

↥S →+* ↥T

The ring homomorphism associated to an inclusion of subrings.

Equations

subring.inclusion h = S.subtype.cod_restrict T _

@[simp]

theorem subring.range_subtype {R : Type u} [ring R] (s : subring R) :

s.subtype.range = s

@[simp]

theorem subring.range_fst {R : Type u} {S : Type v} [ring R] [ring S] :

(ring_hom.fst R S).srange = ⊤

@[simp]

theorem subring.range_snd {R : Type u} {S : Type v} [ring R] [ring S] :

(ring_hom.snd R S).srange = ⊤

@[simp]

theorem subring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [ring R] [ring S] (s : subring R) (t : subring S) :

s.prod ⊥ ⊔ ⊥.prod t = s.prod t

def ring_equiv.subring_congr {R : Type u} [ring R] {s t : subring R} (h : s = t) :

↥s ≃+* ↥t

Makes the identity isomorphism from a proof two subrings of a multiplicative monoid are equal.

Equations

ring_equiv.subring_congr h = {to_fun := (equiv.set_congr _).to_fun, inv_fun := (equiv.set_congr _).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

def ring_equiv.of_left_inverse {R : Type u} {S : Type v} [ring R] [ring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) :

R ≃+* ↥(f.range)

Restrict a ring homomorphism with a left inverse to a ring isomorphism to its ring_hom.range.

Equations

ring_equiv.of_left_inverse h = {to_fun := λ (x : R), ⇑(f.range_restrict) x, inv_fun := λ (x : ↥(f.range)), (g ∘ ⇑(f.range.subtype)) x, left_inv := h, right_inv := _, map_mul' := _, map_add' := _}

@[simp]

theorem ring_equiv.of_left_inverse_apply {R : Type u} {S : Type v} [ring R] [ring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) (x : R) :

↑(⇑(ring_equiv.of_left_inverse h) x) = ⇑f x

@[simp]

theorem ring_equiv.of_left_inverse_symm_apply {R : Type u} {S : Type v} [ring R] [ring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) (x : ↥(f.range)) :

⇑((ring_equiv.of_left_inverse h).symm) x = g ↑x

@[simp]

theorem ring_equiv.subring_map_symm_apply_coe {R : Type u} {S : Type v} [ring R] [ring S] {s : subring R} (e : R ≃+* S) (ᾰ : ↥(add_submonoid.map e.to_add_equiv.to_add_monoid_hom s.to_subsemiring.to_add_submonoid)) :

↑(⇑(e.subring_map.symm) ᾰ) = ⇑(↑↑e.symm) ↑ᾰ

@[simp]

theorem ring_equiv.subring_map_apply_coe {R : Type u} {S : Type v} [ring R] [ring S] {s : subring R} (e : R ≃+* S) (ᾰ : ↥(s.to_subsemiring.to_add_submonoid)) :

↑(⇑(e.subring_map) ᾰ) = ⇑e ↑ᾰ

def ring_equiv.subring_map {R : Type u} {S : Type v} [ring R] [ring S] {s : subring R} (e : R ≃+* S) :

↥s ≃+* ↥(subring.map e.to_ring_hom s)

Given an equivalence e : R ≃+* S of rings and a subring s of R, subring_equiv_map e s is the induced equivalence between s and s.map e

Equations

e.subring_map = e.subsemiring_map s.to_subsemiring

@[protected]

theorem subring.in_closure.rec_on {R : Type u} [ring R] {s : set R} {C : R → Prop} {x : R} (hx : x ∈ subring.closure s) (h1 : C 1) (hneg1 : C (-1)) (hs : ∀ (z : R), z ∈ s → ∀ (n : R), C n → C (z * n)) (ha : ∀ {x y : R}, C x → C y → C (x + y)) :

C x

theorem subring.closure_preimage_le {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set S) :

subring.closure (⇑f ⁻¹' s) ≤ subring.comap f (subring.closure s)

theorem add_subgroup.int_mul_mem {R : Type u} [ring R] {G : add_subgroup R} (k : ℤ) {g : R} (h : g ∈ G) :

↑k * g ∈ G

@[protected, instance]

def subring.has_smul {R : Type u} [ring R] {α : Type u_1} [has_smul R α] (S : subring R) :

has_smul ↥S α

The action by a subring is the action by the underlying ring.

Equations

S.has_smul = S.to_subsemiring.has_smul

theorem subring.smul_def {R : Type u} [ring R] {α : Type u_1} [has_smul R α] {S : subring R} (g : ↥S) (m : α) :

g • m = ↑g • m

@[protected, instance]

def subring.smul_comm_class_left {R : Type u} [ring R] {α : Type u_1} {β : Type u_2} [has_smul R β] [has_smul α β] [smul_comm_class R α β] (S : subring R) :

smul_comm_class ↥S α β

@[protected, instance]

def subring.smul_comm_class_right {R : Type u} [ring R] {α : Type u_1} {β : Type u_2} [has_smul α β] [has_smul R β] [smul_comm_class α R β] (S : subring R) :

smul_comm_class α ↥S β

@[protected, instance]

def subring.is_scalar_tower {R : Type u} [ring R] {α : Type u_1} {β : Type u_2} [has_smul α β] [has_smul R α] [has_smul R β] [is_scalar_tower R α β] (S : subring R) :

is_scalar_tower ↥S α β

Note that this provides is_scalar_tower S R R which is needed by smul_mul_assoc.