Subrings

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)

• instance : CompleteLattice (Subring R) : the complete lattice structure on the subrings.

• Subring.center : the center of a ring R.

• 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 (↑) : Subring M → et M form a GaloisInsertion.

• 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.

• eqLocus 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

theorem Subring.toSubsemiring_strictMono {R : Type u} [Ring R] : StrictMono toSubsemiring

theorem Subring.toSubsemiring_mono {R : Type u} [Ring R] : Monotone toSubsemiring

theorem Subring.toSubsemiring_lt_toSubsemiring {R : Type u} [Ring R] {s t : Subring R} (hst : s < t) : s.toSubsemiring < t.toSubsemiring

theorem Subring.toSubsemiring_le_toSubsemiring {R : Type u} [Ring R] {s t : Subring R} (hst : s ≤ t) : s.toSubsemiring ≤ t.toSubsemiring

theorem Subring.toAddSubgroup_strictMono {R : Type u} [Ring R] : StrictMono toAddSubgroup

theorem Subring.toAddSubgroup_mono {R : Type u} [Ring R] : Monotone toAddSubgroup

theorem Subring.toAddSubgroup_lt_toAddSubgroup {R : Type u} [Ring R] {s t : Subring R} (hst : s < t) : s.toAddSubgroup < t.toAddSubgroup

theorem Subring.toAddSubgroup_le_toAddSubgroup {R : Type u} [Ring R] {s t : Subring R} (hst : s ≤ t) : s.toAddSubgroup ≤ t.toAddSubgroup

theorem Subring.toSubmonoid_strictMono {R : Type u} [Ring R] : StrictMono fun (s : Subring R) => s.toSubmonoid

theorem Subring.toSubmonoid_mono {R : Type u} [Ring R] : Monotone fun (s : Subring R) => s.toSubmonoid

theorem Subring.list_prod_mem {R : Type u} [Ring R] (s : Subring R) {l : List R} : (∀ x ∈ l, x ∈ s) → l.prod ∈ 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} : (∀ x ∈ l, x ∈ s) → l.sum ∈ s

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

theorem Subring.multiset_prod_mem {R : Type u_1} [CommRing R] (s : Subring R) (m : Multiset R) : (∀ a ∈ m, a ∈ s) → m.prod ∈ s

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

theorem Subring.multiset_sum_mem {R : Type u_1} [Ring R] (s : Subring R) (m : Multiset R) : (∀ a ∈ m, a ∈ s) → m.sum ∈ s

Sum of a multiset of elements in a Subring of a Ring is in the Subring.

theorem Subring.prod_mem {R : Type u_1} [CommRing R] (s : Subring R) {ι : Type u_2} {t : Finset ι} {f : ι → R} (h : ∀ c ∈ t, f c ∈ s) : ∏ i ∈ t, f i ∈ s

Product of elements of a subring of a CommRing 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 ∈ t, f c ∈ s) : ∑ i ∈ t, f i ∈ s

Sum of elements in a Subring of a Ring indexed by a Finset is in the Subring.

top

instance Subring.instTop {R : Type u} [Ring R] : Top (Subring R)

The subring R of the ring R.

Equations

• Subring.instTop = { top := let __src := ⊤; let __src_1 := ⊤; { toSubmonoid := __src, 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.topEquiv {R : Type u} [Ring R] : ↥⊤ ≃+* R

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

Equations

• Subring.topEquiv = Subsemiring.topEquiv

@[simp]

theorem Subring.topEquiv_symm_apply_coe {R : Type u} [Ring R] (r : R) : ↑(topEquiv.symm r) = r

@[simp]

theorem Subring.topEquiv_apply {R : Type u} [Ring R] (r : ↥⊤) : topEquiv r = ↑r

instance Subring.instFintypeSubtypeMemTop {R : Type u_1} [Ring R] [Fintype R] : Fintype ↥⊤

Equations

• Subring.instFintypeSubtypeMemTop = inferInstanceAs (Fintype ↑⊤)

theorem Subring.card_top (R : Type u_1) [Ring R] [Fintype R] : Fintype.card ↥⊤ = Fintype.card R

comap

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

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) : ↑(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 ∈ 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) : comap f (comap g s) = 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) : 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) : ↑(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 ∈ map f s ↔ ∃ x ∈ s, f x = y

@[simp]

theorem Subring.map_id {R : Type u} [Ring R] (s : Subring R) : map (RingHom.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) : map g (map f s) = 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} : map f s ≤ t ↔ s ≤ comap f t

theorem Subring.gc_map_comap {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : GaloisConnection (map f) (comap f)

noncomputable def Subring.equivMapOfInjective {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring R) (f : R →+* S) (hf : Function.Injective ⇑f) : ↥s ≃+* ↥(map f s)

A subring is isomorphic to its image under an injective function

Equations

• s.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑s) hf, map_mul' := ⋯, map_add' := ⋯ }

@[simp]

theorem Subring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring R) (f : R →+* S) (hf : Function.Injective ⇑f) (x : ↥s) : ↑((s.equivMapOfInjective f hf) x) = f ↑x

range

def RingHom.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) ⋯

@[simp]

theorem RingHom.coe_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : ↑f.range = Set.range ⇑f

@[simp]

theorem RingHom.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 RingHom.range_eq_map {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : f.range = Subring.map f ⊤

theorem RingHom.mem_range_self {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) (x : R) : f x ∈ f.range

theorem RingHom.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

instance RingHom.fintypeRange {R : Type u} {S : Type v} [Ring R] [Ring S] [Fintype R] [DecidableEq 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.fintypeRange = Set.fintypeRange ⇑f

bot

instance Subring.instBot {R : Type u} [Ring R] : Bot (Subring R)

Equations

• Subring.instBot = { bot := (Int.castRingHom R).range }

instance Subring.instInhabited {R : Type u} [Ring R] : Inhabited (Subring R)

Equations

• Subring.instInhabited = { default := ⊥ }

theorem Subring.coe_bot {R : Type u} [Ring R] : ↑⊥ = Set.range Int.cast

theorem Subring.mem_bot {R : Type u} [Ring R] {x : R} : x ∈ ⊥ ↔ ∃ (n : ℤ), ↑n = x

inf

instance Subring.instMin {R : Type u} [Ring R] : Min (Subring R)

The inf of two subrings is their intersection.

Equations

• One or more equations did not get rendered due to their size.

@[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 Subring.instInfSet {R : Type u} [Ring R] : InfSet (Subring R)

Equations

• Subring.instInfSet = { sInf := fun (s : Set (Subring R)) => Subring.mk' (⋂ t ∈ s, ↑t) (⨅ t ∈ s, t.toSubmonoid) (⨅ t ∈ s, t.toAddSubgroup) ⋯ ⋯ }

@[simp]

theorem Subring.coe_sInf {R : Type u} [Ring R] (S : Set (Subring R)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

theorem Subring.mem_sInf {R : Type u} [Ring R] {S : Set (Subring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

@[simp]

theorem Subring.coe_iInf {R : Type u} [Ring R] {ι : Sort u_1} {S : ι → Subring R} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

theorem Subring.mem_iInf {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.sInf_toSubmonoid {R : Type u} [Ring R] (s : Set (Subring R)) : (sInf s).toSubmonoid = ⨅ t ∈ s, t.toSubmonoid

@[simp]

theorem Subring.sInf_toAddSubgroup {R : Type u} [Ring R] (s : Set (Subring R)) : (sInf s).toAddSubgroup = ⨅ t ∈ s, t.toAddSubgroup

instance Subring.instCompleteLattice {R : Type u} [Ring R] : CompleteLattice (Subring R)

Subrings of a ring form a complete lattice.

Equations

• One or more equations did not get rendered due to their size.

theorem Subring.eq_top_iff' {R : Type u} [Ring R] (A : Subring R) : A = ⊤ ↔ ∀ (x : R), x ∈ A

Center of a ring

def Subring.center (R : Type u) [Ring R] : Subring 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, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

theorem Subring.coe_center (R : Type u) [Ring R] : ↑(center R) = Set.center R

@[simp]

theorem Subring.center_toSubsemiring (R : Type u) [Ring R] : (center R).toSubsemiring = Subsemiring.center R

theorem Subring.mem_center_iff {R : Type u} [Ring R] {z : R} : z ∈ center R ↔ ∀ (g : R), g * z = z * g

instance Subring.decidableMemCenter {R : Type u} [Ring R] [DecidableEq R] [Fintype R] : DecidablePred fun (x : R) => x ∈ center R

Equations

• Subring.decidableMemCenter x✝ = decidable_of_iff' (∀ (g : R), g * x✝ = x✝ * g) ⋯

@[simp]

theorem Subring.center_eq_top (R : Type u_1) [CommRing R] : center R = ⊤

instance Subring.instCommRingSubtypeMemCenter {R : Type u} [Ring R] : CommRing ↥(center R)

The center is commutative.

Equations

• One or more equations did not get rendered due to their size.

def Subring.centerCongr {R : Type u} {S : Type v} [Ring R] [Ring S] (e : R ≃+* S) : ↥(center R) ≃+* ↥(center S)

The center of isomorphic (not necessarily associative) rings are isomorphic.

Equations

• Subring.centerCongr e = NonUnitalSubsemiring.centerCongr e

@[simp]

theorem Subring.centerCongr_symm_apply_coe {R : Type u} {S : Type v} [Ring R] [Ring S] (e : R ≃+* S) (s : ↥(Subsemigroup.center S)) : ↑((centerCongr e).symm s) = (↑e).symm ↑s

@[simp]

theorem Subring.centerCongr_apply_coe {R : Type u} {S : Type v} [Ring R] [Ring S] (e : R ≃+* S) (r : ↥(Subsemigroup.center R)) : ↑((centerCongr e) r) = e ↑r

def Subring.centerToMulOpposite {R : Type u} [Ring R] : ↥(center R) ≃+* ↥(center Rᵐᵒᵖ)

The center of a (not necessarily associative) ring is isomorphic to the center of its opposite.

Equations

• Subring.centerToMulOpposite = NonUnitalSubsemiring.centerToMulOpposite

@[simp]

theorem Subring.centerToMulOpposite_apply_coe {R : Type u} [Ring R] (r : ↥(Subsemigroup.center R)) : ↑(centerToMulOpposite r) = MulOpposite.op ↑r

@[simp]

theorem Subring.centerToMulOpposite_symm_apply_coe {R : Type u} [Ring R] (r : ↥(Subsemigroup.center Rᵐᵒᵖ)) : ↑(centerToMulOpposite.symm r) = MulOpposite.unop ↑r

instance Subring.instField {K : Type u} [DivisionRing K] : Field ↥(center K)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Subring.center.coe_inv {K : Type u} [DivisionRing K] (a : ↥(center K)) : ↑a⁻¹ = (↑a)⁻¹

@[simp]

theorem Subring.center.coe_div {K : Type u} [DivisionRing K] (a b : ↥(center K)) : ↑(a / b) = ↑a / ↑b

def Subring.centralizer {R : Type u} [Ring R] (s : Set R) : Subring R

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

Equations

• Subring.centralizer s = { toSubsemiring := Subsemiring.centralizer s, neg_mem' := ⋯ }

@[simp]

theorem Subring.coe_centralizer {R : Type u} [Ring R] (s : Set R) : ↑(centralizer s) = s.centralizer

theorem Subring.centralizer_toSubmonoid {R : Type u} [Ring R] (s : Set R) : (centralizer s).toSubmonoid = Submonoid.centralizer s

theorem Subring.centralizer_toSubsemiring {R : Type u} [Ring R] (s : Set R) : (centralizer s).toSubsemiring = Subsemiring.centralizer s

theorem Subring.mem_centralizer_iff {R : Type u} [Ring R] {s : Set R} {z : R} : z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g

theorem Subring.center_le_centralizer {R : Type u} [Ring R] (s : Set R) : center R ≤ centralizer s

theorem Subring.centralizer_le {R : Type u} [Ring R] (s t : Set R) (h : s ⊆ t) : centralizer t ≤ centralizer s

@[simp]

theorem Subring.centralizer_eq_top_iff_subset {R : Type u} [Ring R] {s : Set R} : centralizer s = ⊤ ↔ s ⊆ ↑(center R)

@[simp]

theorem Subring.centralizer_univ {R : Type u} [Ring R] : centralizer Set.univ = center R

subring closure of a subset

def Subring.closure {R : Type u} [Ring R] (s : Set R) : Subring R

The Subring generated by a set.

Equations

• Subring.closure s = sInf {S : Subring R | s ⊆ ↑S}

theorem Subring.mem_closure {R : Type u} [Ring R] {x : R} {s : Set R} : x ∈ closure s ↔ ∀ (S : Subring R), s ⊆ ↑S → x ∈ S

@[simp]

theorem Subring.subset_closure {R : Type u} [Ring R] {s : Set R} : s ⊆ ↑(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 ∉ closure s) : P ∉ s

@[simp]

theorem Subring.closure_le {R : Type u} [Ring R] {s : Set R} {t : Subring R} : 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) : closure s ≤ 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 ≤ closure s) : closure s = t

theorem Subring.closure_induction {R : Type u} [Ring R] {s : Set R} {p : (x : R) → x ∈ closure s → Prop} (mem : ∀ (x : R) (hx : x ∈ s), p x ⋯) (zero : p 0 ⋯) (one : p 1 ⋯) (add : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x + y) ⋯) (neg : ∀ (x : R) (hx : x ∈ closure s), p x hx → p (-x) ⋯) (mul : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯) {x : R} (hx : x ∈ closure s) : p x hx

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 : (x y : R) → x ∈ closure s → y ∈ closure s → Prop} (mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯) (zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx) (zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯) (one_left : ∀ (x : R) (hx : x ∈ closure s), p 1 x ⋯ hx) (one_right : ∀ (x : R) (hx : x ∈ closure s), p x 1 hx ⋯) (neg_left : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p (-x) y ⋯ hy) (neg_right : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p x (-y) hx ⋯) (add_left : ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s), p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz) (add_right : ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s), p x y hx hy → p x z hx hz → p x (y + z) hx ⋯) (mul_left : ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s), p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz) (mul_right : ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯) {x y : R} (hx : x ∈ closure s) (hy : y ∈ closure s) : p x y hx hy

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 ∈ closure s ↔ x ∈ AddSubgroup.closure ↑(Submonoid.closure s)

theorem Subring.closure_le_centralizer_centralizer {R : Type u} [Ring R] (s : Set R) : closure s ≤ centralizer ↑(centralizer s)

@[reducible, inline]

abbrev Subring.closureCommRingOfComm {R : Type u} [Ring R] {s : Set R} (hcomm : ∀ x ∈ s, ∀ y ∈ s, x * y = y * x) : CommRing ↥(closure s)

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

Equations

• Subring.closureCommRingOfComm hcomm = { toRing := (Subring.closure s).toRing, mul_comm := ⋯ }

theorem Subring.exists_list_of_mem_closure {R : Type u} [Ring R] {s : Set R} {x : R} (hx : x ∈ closure s) : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = -1) ∧ (List.map List.prod L).sum = x

def Subring.gi (R : Type u) [Ring R] : GaloisInsertion closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• Subring.gi R = { choice := fun (s : Set R) (x : ↑(Subring.closure s) ≤ s) => Subring.closure s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

@[simp]

theorem Subring.closure_eq {R : Type u} [Ring R] (s : Subring R) : closure ↑s = s

Closure of a subring S equals S.

@[simp]

theorem Subring.closure_empty {R : Type u} [Ring R] : closure ∅ = ⊥

@[simp]

theorem Subring.closure_univ {R : Type u} [Ring R] : closure Set.univ = ⊤

theorem Subring.closure_union {R : Type u} [Ring R] (s t : Set R) : closure (s ∪ t) = closure s ⊔ closure t

theorem Subring.closure_iUnion {R : Type u} [Ring R] {ι : Sort u_1} (s : ι → Set R) : closure (⋃ (i : ι), s i) = ⨆ (i : ι), closure (s i)

theorem Subring.closure_sUnion {R : Type u} [Ring R] (s : Set (Set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t

@[simp]

theorem Subring.closure_singleton_intCast {R : Type u} [Ring R] (n : ℤ) : closure {↑n} = ⊥

@[simp]

theorem Subring.closure_singleton_natCast {R : Type u} [Ring R] (n : ℕ) : closure {↑n} = ⊥

@[simp]

theorem Subring.closure_singleton_zero {R : Type u} [Ring R] : closure {0} = ⊥

@[simp]

theorem Subring.closure_singleton_one {R : Type u} [Ring R] : closure {1} = ⊥

@[simp]

theorem Subring.closure_insert_intCast {R : Type u} [Ring R] (n : ℤ) (s : Set R) : closure (insert (↑n) s) = closure s

@[simp]

theorem Subring.closure_insert_natCast {R : Type u} [Ring R] (n : ℕ) (s : Set R) : closure (insert (↑n) s) = closure s

@[simp]

theorem Subring.closure_insert_zero {R : Type u} [Ring R] (s : Set R) : closure (insert 0 s) = closure s

@[simp]

theorem Subring.closure_insert_one {R : Type u} [Ring R] (s : Set R) : closure (insert 1 s) = closure s

theorem Subring.map_sup {R : Type u} {S : Type v} [Ring R] [Ring S] (s t : Subring R) (f : R →+* S) : map f (s ⊔ t) = map f s ⊔ map f t

theorem Subring.map_iSup {R : Type u} {S : Type v} [Ring R] [Ring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subring R) : map f (iSup s) = ⨆ (i : ι), map f (s i)

theorem Subring.map_inf {R : Type u} {S : Type v} [Ring R] [Ring S] (s t : Subring R) (f : R →+* S) (hf : Function.Injective ⇑f) : map f (s ⊓ t) = map f s ⊓ map f t

theorem Subring.map_iInf {R : Type u} {S : Type v} [Ring R] [Ring S] {ι : Sort u_1} [Nonempty ι] (f : R →+* S) (hf : Function.Injective ⇑f) (s : ι → Subring R) : map f (iInf s) = ⨅ (i : ι), 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) : comap f (s ⊓ t) = comap f s ⊓ comap f t

theorem Subring.comap_iInf {R : Type u} {S : Type v} [Ring R] [Ring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subring S) : comap f (iInf s) = ⨅ (i : ι), comap f (s i)

@[simp]

theorem Subring.map_bot {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : map f ⊥ = ⊥

@[simp]

theorem Subring.comap_top {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : 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' := ⋯ }

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.1 ∈ s ∧ p.2 ∈ 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 fun (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 fun (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 ⊤ = comap (RingHom.fst R S) s

theorem Subring.top_prod {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring S) : ⊤.prod s = comap (RingHom.snd R S) s

@[simp]

theorem Subring.top_prod_top {R : Type u} {S : Type v} [Ring R] [Ring S] : ⊤.prod ⊤ = ⊤

def Subring.prodEquiv {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.prodEquiv t = { toEquiv := Equiv.Set.prod ↑s ↑t, map_mul' := ⋯, map_add' := ⋯ }

theorem Subring.mem_iSup_of_directed {R : Type u} [Ring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subring R} (hS : Directed (fun (x1 x2 : Subring R) => x1 ≤ x2) S) {x : R} : x ∈ ⨆ (i : ι), S i ↔ ∃ (i : ι), x ∈ S i

The underlying set of a non-empty directed sSup 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_iSup_of_directed {R : Type u} [Ring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subring R} (hS : Directed (fun (x1 x2 : Subring R) => x1 ≤ x2) S) : ↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem Subring.mem_sSup_of_directedOn {R : Type u} [Ring R] {S : Set (Subring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : Subring R) => x1 ≤ x2) S) {x : R} : x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s

theorem Subring.coe_sSup_of_directedOn {R : Type u} [Ring R] {S : Set (Subring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : Subring R) => x1 ≤ x2) S) : ↑(sSup S) = ⋃ 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 ∈ 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) : map (↑f) K = 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) : comap (↑f) K = map (↑f.symm) K

def RingHom.rangeRestrict {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.rangeFactorization.

Equations

• f.rangeRestrict = f.codRestrict f.range ⋯

@[simp]

theorem RingHom.coe_rangeRestrict {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) (x : R) : ↑(f.rangeRestrict x) = f x

theorem RingHom.rangeRestrict_surjective {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : Function.Surjective ⇑f.rangeRestrict

theorem RingHom.range_eq_top {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} : f.range = ⊤ ↔ Function.Surjective ⇑f

@[deprecated RingHom.range_eq_top (since := "2024-11-11")]

theorem RingHom.range_top_iff_surjective {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} : f.range = ⊤ ↔ Function.Surjective ⇑f

Alias of RingHom.range_eq_top.

@[simp]

theorem RingHom.range_eq_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.

@[deprecated RingHom.range_eq_top_of_surjective (since := "2024-11-11")]

theorem RingHom.range_top_of_surjective {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) (hf : Function.Surjective ⇑f) : f.range = ⊤

Alias of RingHom.range_eq_top_of_surjective.

---

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

def RingHom.eqLocus {R : Type u} [Ring R] {S : Type v} [Semiring S] (f g : R →+* S) : Subring R

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.eqLocus g = { carrier := {x : R | f x = g x}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

@[simp]

theorem RingHom.eqLocus_same {R : Type u} [Ring R] {S : Type v} [Semiring S] (f : R →+* S) : f.eqLocus f = ⊤

theorem RingHom.eqOn_set_closure {R : Type u} [Ring R] {S : Type v} [Semiring S] {f g : R →+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subring.closure s)

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

theorem RingHom.eq_of_eqOn_set_top {R : Type u} [Ring R] {S : Type v} [Semiring S] {f g : R →+* S} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g

theorem RingHom.eq_of_eqOn_set_dense {R : Type u} [Ring R] {S : Type v} [Semiring S] {s : Set R} (hs : Subring.closure s = ⊤) {f g : R →+* S} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem RingHom.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 RingHom.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.codRestrict T ⋯

@[simp]

theorem Subring.range_subtype {R : Type u} [Ring R] (s : Subring R) : s.subtype.range = s

theorem Subring.range_fst {R : Type u} {S : Type v} [Ring R] [Ring S] : (RingHom.fst R S).rangeS = ⊤

theorem Subring.range_snd {R : Type u} {S : Type v} [Ring R] [Ring S] : (RingHom.snd R S).rangeS = ⊤

@[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 RingEquiv.subringCongr {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

• RingEquiv.subringCongr h = { toEquiv := Equiv.setCongr ⋯, map_mul' := ⋯, map_add' := ⋯ }

def RingEquiv.ofLeftInverse {R : Type u} {S : Type v} [Ring R] [Ring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) : R ≃+* ↥f.range

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

Equations

• RingEquiv.ofLeftInverse h = { toFun := fun (x : R) => f.rangeRestrict x, invFun := fun (x : ↥f.range) => (g ∘ ⇑f.range.subtype) x, left_inv := h, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯ }

@[simp]

theorem RingEquiv.ofLeftInverse_apply {R : Type u} {S : Type v} [Ring R] [Ring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) (x : R) : ↑((ofLeftInverse h) x) = f x

@[simp]

theorem RingEquiv.ofLeftInverse_symm_apply {R : Type u} {S : Type v} [Ring R] [Ring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) (x : ↥f.range) : (ofLeftInverse h).symm x = g ↑x

def RingEquiv.subringMap {R : Type u} {S : Type v} [Ring R] [Ring S] {s : Subring R} (e : R ≃+* S) : ↥s ≃+* ↥(Subring.map e.toRingHom s)

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

Equations

• e.subringMap = e.subsemiringMap s.toSubsemiring

theorem Subring.InClosure.recOn {R : Type u} [Ring R] {s : Set R} {C : R → Prop} {x : R} (hx : x ∈ closure s) (h1 : C 1) (hneg1 : C (-1)) (hs : ∀ 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) : closure (⇑f ⁻¹' s) ≤ comap f (closure s)

Actions by Subrings

These are just copies of the definitions about Subsemiring starting from Subsemiring.MulAction.

When R is commutative, Algebra.ofSubring provides a stronger result than those found in this file, which uses the same scalar action.

instance Subring.instSMulSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [SMul R α] (S : Subring R) : SMul (↥S) α

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

Equations

• S.instSMulSubtypeMem = inferInstanceAs (SMul (↥S.toSubsemiring) α)

theorem Subring.smul_def {R : Type u} [Ring R] {α : Type u_1} [SMul R α] {S : Subring R} (g : ↥S) (m : α) : g • m = ↑g • m

instance Subring.smulCommClass_left {R : Type u} [Ring R] {α : Type u_1} {β : Type u_2} [SMul R β] [SMul α β] [SMulCommClass R α β] (S : Subring R) : SMulCommClass (↥S) α β

instance Subring.smulCommClass_right {R : Type u} [Ring R] {α : Type u_1} {β : Type u_2} [SMul α β] [SMul R β] [SMulCommClass α R β] (S : Subring R) : SMulCommClass α (↥S) β

instance Subring.instIsScalarTowerSubtypeMem {R : Type u} [Ring R] {α : Type u_1} {β : Type u_2} [SMul α β] [SMul R α] [SMul R β] [IsScalarTower R α β] (S : Subring R) : IsScalarTower (↥S) α β

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

instance Subring.instFaithfulSMulSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [SMul R α] [FaithfulSMul R α] (S : Subring R) : FaithfulSMul (↥S) α

instance Subring.instMulActionSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [MulAction R α] (S : Subring R) : MulAction (↥S) α

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

Equations

• S.instMulActionSubtypeMem = inferInstanceAs (MulAction (↥S.toSubsemiring) α)

instance Subring.instDistribMulActionSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [AddMonoid α] [DistribMulAction R α] (S : Subring R) : DistribMulAction (↥S) α

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

Equations

• S.instDistribMulActionSubtypeMem = inferInstanceAs (DistribMulAction (↥S.toSubsemiring) α)

instance Subring.instMulDistribMulActionSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [Monoid α] [MulDistribMulAction R α] (S : Subring R) : MulDistribMulAction (↥S) α

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

Equations

• S.instMulDistribMulActionSubtypeMem = inferInstanceAs (MulDistribMulAction (↥S.toSubsemiring) α)

instance Subring.instSMulWithZeroSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [Zero α] [SMulWithZero R α] (S : Subring R) : SMulWithZero (↥S) α

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

Equations

• S.instSMulWithZeroSubtypeMem = inferInstanceAs (SMulWithZero (↥S.toSubsemiring) α)

instance Subring.instMulActionWithZeroSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [Zero α] [MulActionWithZero R α] (S : Subring R) : MulActionWithZero (↥S) α

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

Equations

• S.instMulActionWithZeroSubtypeMem = S.mulActionWithZero

instance Subring.instModuleSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [AddCommMonoid α] [Module R α] (S : Subring R) : Module (↥S) α

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

Equations

• S.instModuleSubtypeMem = S.module

instance Subring.instMulSemiringActionSubtypeMem {R : Type u} [Ring R] {α : Type u_1} [Semiring α] [MulSemiringAction R α] (S : Subring R) : MulSemiringAction (↥S) α

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

Equations

• S.instMulSemiringActionSubtypeMem = inferInstanceAs (MulSemiringAction (↥S.toSubmonoid) α)

instance Subring.center.smulCommClass_left {R : Type u} [Ring R] : SMulCommClass (↥(center R)) R R

The center of a semiring acts commutatively on that semiring.

instance Subring.center.smulCommClass_right {R : Type u} [Ring R] : SMulCommClass R (↥(center R)) R

The center of a semiring acts commutatively on that semiring.

theorem Subring.map_comap_eq {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) (t : Subring S) : map f (comap f t) = t ⊓ f.range

theorem Subring.map_comap_eq_self {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} {t : Subring S} (h : t ≤ f.range) : map f (comap f t) = t

theorem Subring.map_comap_eq_self_of_surjective {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (t : Subring S) : map f (comap f t) = t

theorem Subring.comap_map_eq {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) (s : Subring R) : comap f (map f s) = s ⊔ closure (⇑f ⁻¹' {0})

theorem Subring.comap_map_eq_self {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} {s : Subring R} (h : ⇑f ⁻¹' {0} ⊆ ↑s) : comap f (map f s) = s

theorem Subring.comap_map_eq_self_of_injective {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} (hf : Function.Injective ⇑f) (s : Subring R) : comap f (map f s) = s

theorem AddSubgroup.int_mul_mem {R : Type u} [Ring R] {G : AddSubgroup R} (k : ℤ) {g : R} (h : g ∈ G) : ↑k * g ∈ G