mathlib documentation

ring_theory.subring

Subrings

Let R be a ring. This file defines the "bundled" subring type subring R, a type whose terms correspond to subrings of R. This is the preferred way to talk about subrings in mathlib. Unbundled subrings (s : set R and is_subring s) are not in this file, and they will ultimately be deprecated.

We prove that subrings are a complete lattice, and that you can map (pushforward) and comap (pull back) them along ring homomorphisms.

We define the closure construction from set R to subring R, sending a subset of R to the subring it generates, and prove that it is a Galois insertion.

Main definitions

Notation used here:

(R : Type u) [ring R] (S : Type u) [ring S] (f g : R →+* S) (A : subring R) (B : subring S) (s : set R)

subring R : the type of subrings of a ring R.
instance : complete_lattice (subring R) : the complete lattice structure on the subrings.
subring.closure : subring closure of a set, i.e., the smallest subring that includes the set.
subring.gi : closure : set M → subring M and coercion coe : subring M → set M form a galois_insertion.
comap f B : subring A : the preimage of a subring B along the ring homomorphism f
map f A : subring B : the image of a subring A along the ring homomorphism f.
prod A B : subring (R × S) : the product of subrings
f.range : subring B : the range of the ring homomorphism f.
eq_locus f g : subring R : given ring homomorphisms f g : R →+* S, the subring of R where f x = g x

Implementation notes

A subring is implemented as a subsemiring which is also an additive subgroup. The initial PR was as a submonoid which is also an additive subgroup.

Lattice inclusion (e.g. ≤ and ⊓) is used rather than set notation (⊆ and ∩), although ∈ is defined as membership of a subring's underlying set.

Tags

subring, subrings

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

Reinterpret a subring as an add_subgroup.

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

Type u

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

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

Reinterpret a subring as a subsemiring.

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

The underlying submonoid of a subring.

Equations

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

@[instance]

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

has_coe (subring R) (set R)

Equations

subring.has_coe = {coe := subring.carrier _inst_1}

@[instance]

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

has_coe_to_sort (subring R)

Equations

subring.has_coe_to_sort = {S := Type u, coe := λ (S : subring R), ↥(S.carrier)}

@[instance]

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

has_mem R (subring R)

Equations

subring.has_mem = {mem := λ (m : R) (S : subring R), m ∈ ↑S}

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, one_mem' := _, mul_mem' := _, zero_mem' := _, add_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 set.to_subring {R : Type u} [ring R] (S : set R) [is_subring S] :

Construct a subring from a set satisfying is_subring.

Equations

S.to_subring = {carrier := S, one_mem' := _, mul_mem' := _, zero_mem' := _, add_mem' := _, neg_mem' := _}

theorem subring.exists {R : Type u} [ring R] {s : subring R} {p : ↥s → Prop} :

(∃ (x : ↥s), p x) ↔ ∃ (x : R) (H : x ∈ s), p ⟨x, H⟩

theorem subring.forall {R : Type u} [ring R] {s : subring R} {p : ↥s → Prop} :

(∀ (x : ↥s), p x) ↔ ∀ (x : R) (H : x ∈ s), p ⟨x, H⟩

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, one_mem' := _, mul_mem' := _, zero_mem' := _, add_mem' := _, neg_mem' := _}

theorem subring.ext' {R : Type u} [ring R] ⦃s t : subring R⦄ (h : ↑s = ↑t) :

s = t

Two subrings are equal if the underlying subsets are equal.

theorem subring.ext'_iff {R : Type u} [ring R] {s t : subring R} :

s = t ↔ ↑s = ↑t

Two subrings are equal if and only if the underlying subsets are equal.

@[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.

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

1 ∈ s

A subring contains the ring's 1.

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

0 ∈ s

A subring contains the ring's 0.

theorem subring.mul_mem {R : Type u} [ring R] (s : subring R) {x y : R} (a : x ∈ s) (a_1 : y ∈ s) :

x * y ∈ s

A subring is closed under multiplication.

theorem subring.add_mem {R : Type u} [ring R] (s : subring R) {x y : R} (a : x ∈ s) (a_1 : y ∈ s) :

x + y ∈ s

A subring is closed under addition.

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

-x ∈ s

A subring is closed under negation.

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

theorem subring.list_prod_mem {R : Type u} [ring R] (s : subring R) {l : list R} (a : ∀ (x : R), x ∈ l → x ∈ s) :

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

theorem subring.list_sum_mem {R : Type u} [ring R] (s : subring R) {l : list R} (a : ∀ (x : R), x ∈ l → x ∈ s) :

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

theorem subring.multiset_prod_mem {R : Type u_1} [comm_ring R] (s : subring R) (m : multiset R) (a : ∀ (a : R), a ∈ m → a ∈ s) :

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

theorem subring.multiset_sum_mem {R : Type u_1} [ring R] (s : subring R) (m : multiset R) (a : ∀ (a : R), a ∈ m → a ∈ s) :

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

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) :

∏ (i : ι) in t, f i ∈ s

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

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) :

∑ (i : ι) in t, f i ∈ s

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

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

x ^ n ∈ s

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

n •ℤ x ∈ s

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

@[instance]

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

A subring of a ring inherits a ring structure

Equations

s.to_ring = {add := add_comm_group.add s.to_add_subgroup.to_add_comm_group, add_assoc := _, zero := add_comm_group.zero s.to_add_subgroup.to_add_comm_group, zero_add := _, add_zero := _, neg := add_comm_group.neg s.to_add_subgroup.to_add_comm_group, add_left_neg := _, add_comm := _, mul := monoid.mul s.to_submonoid.to_monoid, mul_assoc := _, one := monoid.one s.to_submonoid.to_monoid, one_mul := _, mul_one := _, left_distrib := _, right_distrib := _}

@[simp]

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

↑(x + y) = ↑x + ↑y

@[simp]

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

@[simp]

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

↑x * y = (↑x) * ↑y

@[simp]

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

↑0 = 0

@[simp]

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

↑1 = 1

@[simp]

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

↑x = 0 ↔ x = 0

@[instance]

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

A subring of a comm_ring is a comm_ring.

Equations

s.to_comm_ring = {add := ring.add s.to_ring, add_assoc := _, zero := ring.zero s.to_ring, zero_add := _, add_zero := _, neg := ring.neg s.to_ring, add_left_neg := _, add_comm := _, mul := ring.mul s.to_ring, mul_assoc := _, one := ring.one s.to_ring, one_mul := _, mul_one := _, left_distrib := _, right_distrib := _, mul_comm := _}

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

Partial order

@[instance]

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

partial_order (subring R)

Equations

subring.partial_order = {le := λ (s t : subring R), ∀ ⦃x : R⦄, x ∈ s → x ∈ t, lt := partial_order.lt (partial_order.lift coe subring.ext'), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

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

s ≤ t ↔ ∀ ⦃x : R⦄, x ∈ s → x ∈ t

@[simp]

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

↑s ⊆ ↑t ↔ s ≤ t

@[simp]

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

↑s ⊂ ↑t ↔ s < t

@[simp]

theorem subring.mem_coe {R : Type u} [ring R] {S : subring R} {m : R} :

m ∈ ↑S ↔ m ∈ S

@[simp]

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

@[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

top

@[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, one_mem' := _, mul_mem' := _, zero_mem' := _, add_mem' := _, neg_mem' := _}}

@[simp]

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

@[simp]

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

↑⊤ = set.univ

comap

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, one_mem' := _, mul_mem' := _, zero_mem' := _, add_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

map

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

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

Equations

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

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)

range

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

The range of a ring homomorphism, as a subring of the target.

Equations

f.range = subring.map f ⊤

@[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.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

def ring_hom.cod_restrict' {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : subring S) (h : ∀ (x : R), ⇑f x ∈ s) :

Restrict the codomain of a ring homomorphism to a subring that includes the range.

Equations

f.cod_restrict' s h = {to_fun := λ (x : R), ⟨⇑f x, _⟩, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

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

function.surjective ⇑(f.cod_restrict' f.range _)

def subring.subset_comm_ring {cR : Type u} [comm_ring cR] (S : subring cR) :

A subring of a commutative ring is a commutative ring.

Equations

S.subset_comm_ring = {add := ring.add S.to_ring, add_assoc := _, zero := ring.zero S.to_ring, zero_add := _, add_zero := _, neg := ring.neg S.to_ring, add_left_neg := _, add_comm := _, mul := ring.mul S.to_ring, mul_assoc := _, one := ring.one S.to_ring, one_mul := _, mul_one := _, left_distrib := _, right_distrib := _, mul_comm := _}

@[instance]

def subring.subring.domain {D : Type u_1} [integral_domain D] (S : subring D) :

integral_domain ↥S

A subring of an integral domain is an integral domain.

Equations

subring.subring.domain S = {add := comm_ring.add S.subset_comm_ring, add_assoc := _, zero := comm_ring.zero S.subset_comm_ring, zero_add := _, add_zero := _, neg := comm_ring.neg S.subset_comm_ring, add_left_neg := _, add_comm := _, mul := comm_ring.mul S.subset_comm_ring, mul_assoc := _, one := comm_ring.one S.subset_comm_ring, one_mul := _, mul_one := _, left_distrib := _, right_distrib := _, mul_comm := _, exists_pair_ne := _, eq_zero_or_eq_zero_of_mul_eq_zero := _}

bot

@[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}

@[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

inf

@[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, one_mem' := _, mul_mem' := _, zero_mem' := _, add_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'

@[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]

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

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

@[simp]

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

(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)) :

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

@[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 := _, top := ⊤, le_top := _, bot := ⊥, bot_le := _, Sup := complete_lattice.Sup (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), Inf := complete_lattice.Inf (complete_lattice_of_Inf (subring R) subring.complete_lattice._proof_1), le_Sup := _, Sup_le := _, Inf_le := _, le_Inf := _}

subring closure of a subset

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

The subring generated by a set.

Equations

subring.closure s = 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.

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

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

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

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

galois_insertion subring.closure coe

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 ∅ = ⊥

@[simp]

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

subring.closure set.univ = ⊤

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 (⋃₀s) = ⨆ (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.prod ↑t, one_mem' := _, mul_mem' := _, zero_mem' := _, add_mem' := _, neg_mem' := _}

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

↑(s.prod t) = ↑s.prod ↑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 ∈ 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) :

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

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

Restriction of a ring homomorphism to a subring of the domain.

Equations

f.restrict s = f.comp s.subtype

@[simp]

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

⇑(f.restrict s) x = ⇑f ↑x

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.

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_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) :

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}, one_mem' := _, mul_mem' := _, zero_mem' := _, add_mem' := _, neg_mem' := _}

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) :

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' := _}

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)