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 IsSubring 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 : 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

class SubringClass (S : Type u_1) (R : Type u) [Ring R] [SetLike S R] extends SubsemiringClass , NegMemClass : Prop

SubringClass S R states that S is a type of subsets s ⊆ R that are both a multiplicative submonoid and an additive subgroup.

instance SubringClass.addSubgroupClass (S : Type u_1) (R : Type u) [SetLike S R] [Ring R] [h : SubringClass S R] : AddSubgroupClass S R

Equations

• SubringClass.addSubgroupClass = SubringClass.addSubgroupClass.proof_1

theorem coe_int_mem {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) (n : ℤ) : ↑n ∈ s

instance SubringClass.toHasIntCast {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) : IntCast { x // x ∈ s }

Equations

• SubringClass.toHasIntCast s = { intCast := fun n => { val := ↑n, property := (_ : ↑n ∈ s) } }

instance SubringClass.toRing {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) : Ring { x // x ∈ s }

A subring of a ring inherits a ring structure

Equations

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

instance SubringClass.toCommRing {S : Type v} (s : S) {R : Type u_1} [CommRing R] [SetLike S R] [SubringClass S R] : CommRing { x // x ∈ s }

A subring of a CommRing is a CommRing.

Equations

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

instance SubringClass.instIsDomainSubtypeMemInstMembershipToSemiringToRing {S : Type v} (s : S) {R : Type u_1} [Ring R] [IsDomain R] [SetLike S R] [SubringClass S R] : IsDomain { x // x ∈ s }

A subring of a domain is a domain.

Equations

• @SubringClass.instIsDomainSubtypeMemInstMembershipToSemiringToRing = @SubringClass.instIsDomainSubtypeMemInstMembershipToSemiringToRing.proof_1

instance SubringClass.toOrderedRing {S : Type v} (s : S) {R : Type u_1} [OrderedRing R] [SetLike S R] [SubringClass S R] : OrderedRing { x // x ∈ s }

A subring of an OrderedRing is an OrderedRing.

Equations

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

instance SubringClass.toOrderedCommRing {S : Type v} (s : S) {R : Type u_1} [OrderedCommRing R] [SetLike S R] [SubringClass S R] : OrderedCommRing { x // x ∈ s }

A subring of an OrderedCommRing is an OrderedCommRing.

Equations

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

instance SubringClass.toLinearOrderedRing {S : Type v} (s : S) {R : Type u_1} [LinearOrderedRing R] [SetLike S R] [SubringClass S R] : LinearOrderedRing { x // x ∈ s }

A subring of a LinearOrderedRing is a LinearOrderedRing.

Equations

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

instance SubringClass.toLinearOrderedCommRing {S : Type v} (s : S) {R : Type u_1} [LinearOrderedCommRing R] [SetLike S R] [SubringClass S R] : LinearOrderedCommRing { x // x ∈ s }

A subring of a LinearOrderedCommRing is a LinearOrderedCommRing.

Equations

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

def SubringClass.subtype {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) : { x // x ∈ s } →+* R

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

Equations

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

@[simp]

theorem SubringClass.coeSubtype {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) : ↑(SubringClass.subtype s) = Subtype.val

@[simp]

theorem SubringClass.coe_natCast {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) (n : ℕ) : ↑↑n = ↑n

@[simp]

theorem SubringClass.coe_intCast {R : Type u} {S : Type v} [Ring R] [SetLike S R] [hSR : SubringClass S R] (s : S) (n : ℤ) : ↑↑n = ↑n

structure Subring (R : Type u) [Ring R] extends Subsemiring : Type u

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

  G is closed under negation

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.

@[reducible]

abbrev Subring.toAddSubgroup {R : Type u} [Ring R] (self : Subring R) : AddSubgroup R

Reinterpret a Subring as an AddSubgroup.

Equations

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

instance Subring.instSetLikeSubring {R : Type u} [Ring R] : SetLike (Subring R) R

Equations

• Subring.instSetLikeSubring = { coe := fun s => s.carrier, coe_injective' := (_ : ∀ (p q : Subring R), (fun s => s.carrier) p = (fun s => s.carrier) q → p = q) }

instance Subring.instSubringClassSubringInstSetLikeSubring {R : Type u} [Ring R] : SubringClass (Subring R) R

Equations

• @Subring.instSubringClassSubringInstSetLikeSubring = @Subring.instSubringClassSubringInstSetLikeSubring.proof_1

@[simp]

theorem Subring.mem_toSubsemiring {R : Type u} [Ring R] {s : Subring R} {x : R} : x ∈ s.toSubsemiring ↔ x ∈ s

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 : Subsemiring R} {x : R} (h : ∀ {x : R}, x ∈ S.carrier → -x ∈ S.carrier) : x ∈ { toSubsemiring := S, neg_mem' := h } ↔ x ∈ S

@[simp]

theorem Subring.coe_set_mk {R : Type u} [Ring R] (S : Subsemiring R) (h : ∀ {x : R}, x ∈ S.carrier → -x ∈ S.carrier) : ↑{ toSubsemiring := S, neg_mem' := h } = ↑S

@[simp]

theorem Subring.mk_le_mk {R : Type u} [Ring R] {S : Subsemiring R} {S' : Subsemiring R} (h₁ : ∀ {x : R}, x ∈ S.carrier → -x ∈ S.carrier) (h₂ : ∀ {x : R}, x ∈ S'.carrier → -x ∈ S'.carrier) : { toSubsemiring := S, neg_mem' := h₁ } ≤ { toSubsemiring := S', neg_mem' := h₂ } ↔ S ≤ S'

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

Two subrings are equal if they have the same elements.

def Subring.copy {R : Type u} [Ring R] (S : Subring R) (s : Set R) (hs : s = ↑S) : Subring R

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

Equations

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

@[simp]

theorem Subring.coe_copy {R : Type u} [Ring R] (S : Subring R) (s : Set R) (hs : s = ↑S) : ↑(Subring.copy S s hs) = s

theorem Subring.copy_eq {R : Type u} [Ring R] (S : Subring R) (s : Set R) (hs : s = ↑S) : Subring.copy S s hs = S

theorem Subring.toSubsemiring_injective {R : Type u} [Ring R] : Function.Injective Subring.toSubsemiring

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

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

theorem Subring.toAddSubgroup_injective {R : Type u} [Ring R] : Function.Injective Subring.toAddSubgroup

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

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

theorem Subring.toSubmonoid_injective {R : Type u} [Ring R] : Function.Injective fun s => s.toSubmonoid

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

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

def Subring.mk' {R : Type u} [Ring R] (s : Set R) (sm : Submonoid R) (sa : AddSubgroup R) (hm : ↑sm = s) (ha : ↑sa = s) : Subring R

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

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

@[simp]

theorem Subring.coe_mk' {R : Type u} [Ring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubgroup 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 : AddSubgroup R} (ha : ↑sa = s) {x : R} : x ∈ Subring.mk' s sm sa hm ha ↔ x ∈ s

@[simp]

theorem Subring.mk'_toSubmonoid {R : Type u} [Ring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubgroup R} (ha : ↑sa = s) : (Subring.mk' s sm sa hm ha).toSubsemiring.toSubmonoid = sm

@[simp]

theorem Subring.mk'_toAddSubgroup {R : Type u} [Ring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubgroup R} (ha : ↑sa = s) : Subring.toAddSubgroup (Subring.mk' s sm sa hm ha) = sa

def Subsemiring.toSubring {R : Type u} [Ring R] (s : Subsemiring R) (hneg : -1 ∈ s) : Subring R

A Subsemiring containing -1 is a Subring.

Equations

• Subsemiring.toSubring s hneg = { toSubsemiring := s, neg_mem' := (_ : ∀ {x : R}, x ∈ s.carrier → -x ∈ s.carrier) }

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 : R} {y : R} : x ∈ s → y ∈ s → x * y ∈ s

A subring is closed under multiplication.

theorem Subring.add_mem {R : Type u} [Ring R] (s : Subring R) {x : R} {y : R} : x ∈ s → 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} : x ∈ s → -x ∈ s

A subring is closed under negation.

theorem Subring.sub_mem {R : Type u} [Ring R] (s : Subring R) {x : R} {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} : (∀ (x : R), x ∈ l → x ∈ s) → List.prod l ∈ 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 : R), x ∈ l → x ∈ s) → List.sum l ∈ 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 : R), a ∈ m → a ∈ s) → Multiset.prod m ∈ 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 : R), a ∈ m → a ∈ s) → Multiset.sum m ∈ 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 : ι), c ∈ t → f c ∈ s) : (Finset.prod t fun i => 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 : ι), c ∈ t → f c ∈ s) : (Finset.sum t fun i => f i) ∈ s

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

instance Subring.toRing {R : Type u} [Ring R] (s : Subring R) : Ring { x // x ∈ s }

A subring of a ring inherits a ring structure

Equations

• Subring.toRing s = SubringClass.toRing s

theorem Subring.zsmul_mem {R : Type u} [Ring R] (s : Subring R) {x : R} (hx : x ∈ s) (n : ℤ) : n • x ∈ s

theorem Subring.pow_mem {R : Type u} [Ring R] (s : Subring R) {x : R} (hx : x ∈ s) (n : ℕ) : x ^ n ∈ s

@[simp]

theorem Subring.coe_add {R : Type u} [Ring R] (s : Subring R) (x : { x // x ∈ s }) (y : { x // x ∈ s }) : ↑(x + y) = ↑x + ↑y

@[simp]

theorem Subring.coe_neg {R : Type u} [Ring R] (s : Subring R) (x : { x // x ∈ s }) : ↑(-x) = -↑x

@[simp]

theorem Subring.coe_mul {R : Type u} [Ring R] (s : Subring R) (x : { x // x ∈ s }) (y : { x // x ∈ 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_pow {R : Type u} [Ring R] (s : Subring R) (x : { x // x ∈ s }) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

theorem Subring.coe_eq_zero_iff {R : Type u} [Ring R] (s : Subring R) {x : { x // x ∈ s }} : ↑x = 0 ↔ x = 0

instance Subring.toCommRing {R : Type u_1} [CommRing R] (s : Subring R) : CommRing { x // x ∈ s }

A subring of a CommRing is a CommRing.

Equations

• Subring.toCommRing s = SubringClass.toCommRing s

instance Subring.instNontrivialSubtypeMemSubringInstMembershipInstSetLikeSubring {R : Type u_1} [Ring R] [Nontrivial R] (s : Subring R) : Nontrivial { x // x ∈ s }

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

Equations

• @Subring.instNontrivialSubtypeMemSubringInstMembershipInstSetLikeSubring = @Subring.instNontrivialSubtypeMemSubringInstMembershipInstSetLikeSubring.proof_1

instance Subring.instNoZeroDivisorsSubtypeMemSubringInstMembershipInstSetLikeSubringMulToMulOneClassToMulZeroOneClassToNonAssocSemiringToSemiringToSubmonoidToSubsemiringZeroToZeroToMonoidWithZeroToZeroMemClassToAddZeroClassToAddMonoidToAddMonoidWithOneToAddGroupWithOneToAddSubmonoidClassToSubsemiringClassInstSubringClassSubringInstSetLikeSubring {R : Type u_1} [Ring R] [NoZeroDivisors R] (s : Subring R) : NoZeroDivisors { x // x ∈ s }

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

Equations

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

instance Subring.instIsDomainSubtypeMemSubringInstMembershipInstSetLikeSubringToSemiringToSemiringToSubsemiring {R : Type u_1} [Ring R] [IsDomain R] (s : Subring R) : IsDomain { x // x ∈ s }

A subring of a domain is a domain.

Equations

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

instance Subring.toOrderedRing {R : Type u_1} [OrderedRing R] (s : Subring R) : OrderedRing { x // x ∈ s }

A subring of an OrderedRing is an OrderedRing.

Equations

• Subring.toOrderedRing s = SubringClass.toOrderedRing s

instance Subring.toOrderedCommRing {R : Type u_1} [OrderedCommRing R] (s : Subring R) : OrderedCommRing { x // x ∈ s }

A subring of an OrderedCommRing is an OrderedCommRing.

Equations

• Subring.toOrderedCommRing s = SubringClass.toOrderedCommRing s

instance Subring.toLinearOrderedRing {R : Type u_1} [LinearOrderedRing R] (s : Subring R) : LinearOrderedRing { x // x ∈ s }

A subring of a LinearOrderedRing is a LinearOrderedRing.

Equations

• Subring.toLinearOrderedRing s = SubringClass.toLinearOrderedRing s

instance Subring.toLinearOrderedCommRing {R : Type u_1} [LinearOrderedCommRing R] (s : Subring R) : LinearOrderedCommRing { x // x ∈ s }

A subring of a LinearOrderedCommRing is a LinearOrderedCommRing.

Equations

• Subring.toLinearOrderedCommRing s = SubringClass.toLinearOrderedCommRing s

def Subring.subtype {R : Type u} [Ring R] (s : Subring R) : { x // x ∈ s } →+* R

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

Equations

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

@[simp]

theorem Subring.coeSubtype {R : Type u} [Ring R] (s : Subring R) : ↑(Subring.subtype s) = Subtype.val

theorem Subring.coe_natCast {R : Type u} [Ring R] (s : Subring R) (n : ℕ) : ↑↑n = ↑n

theorem Subring.coe_intCast {R : Type u} [Ring R] (s : Subring R) (n : ℤ) : ↑↑n = ↑n

Partial order

@[simp]

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

@[simp]

theorem Subring.mem_toSubmonoid {R : Type u} [Ring R] {s : Subring R} {x : R} : x ∈ s.toSubmonoid ↔ x ∈ s

@[simp]

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

@[simp]

theorem Subring.mem_toAddSubgroup {R : Type u} [Ring R] {s : Subring R} {x : R} : x ∈ Subring.toAddSubgroup s ↔ x ∈ s

@[simp]

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

top

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

The subring R of the ring R.

Equations

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

@[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

@[simp]

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

@[simp]

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

def Subring.topEquiv {R : Type u} [Ring R] : { x // x ∈ ⊤ } ≃+* R

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

Equations

• Subring.topEquiv = Subsemiring.topEquiv

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

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

@[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 (RingHom.comp g 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

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

@[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, x ∈ s ∧ ↑f x = y

@[simp]

theorem Subring.map_id {R : Type u} [Ring R] (s : Subring R) : Subring.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) : Subring.map g (Subring.map f s) = Subring.map (RingHom.comp g 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) : GaloisConnection (Subring.map f) (Subring.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) : { x // x ∈ s } ≃+* { x // x ∈ Subring.map f s }

A subring is isomorphic to its image under an injective function

Equations

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

@[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 : { x // x ∈ s }) : ↑(↑(Subring.equivMapOfInjective s 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

• RingHom.range f = Subring.copy (Subring.map f ⊤) (Set.range ↑f) (_ : Set.range ↑f = ↑f '' Set.univ)

@[simp]

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

@[simp]

theorem RingHom.mem_range {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} {y : S} : y ∈ RingHom.range f ↔ ∃ x, ↑f x = y

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

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 (RingHom.range f) = RingHom.range (RingHom.comp g f)

instance RingHom.fintypeRange {R : Type u} {S : Type v} [Ring R] [Ring S] [Fintype R] [DecidableEq S] (f : R →+* S) : Fintype { x // x ∈ RingHom.range f }

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

• RingHom.fintypeRange f = Set.fintypeRange ↑f

bot

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

Equations

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

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

Equations

• Subring.instInhabitedSubring = { 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, ↑n = x

inf

instance Subring.instInfSubring {R : Type u} [Ring R] : Inf (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 : Subring R) (p' : Subring R) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem Subring.mem_inf {R : Type u} [Ring R] {p : Subring R} {p' : Subring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

instance Subring.instInfSetSubring {R : Type u} [Ring R] : InfSet (Subring R)

Equations

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

@[simp]

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

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

@[simp]

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

instance Subring.instCompleteLatticeSubring {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

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

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

@[simp]

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

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

Equations

• Subring.decidableMemCenter x = decidable_of_iff' (∀ (g : R), g * x = x * g) (_ : x ∈ Subring.center R ↔ ∀ (g : R), g * x = x * g)

@[simp]

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

instance Subring.instCommRingSubtypeMemSubringInstMembershipInstSetLikeSubringCenter {R : Type u} [Ring R] : CommRing { x // x ∈ Subring.center R }

The center is commutative.

Equations

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

instance Subring.instFieldSubtypeMemSubringToRingInstMembershipInstSetLikeSubringCenter {K : Type u} [DivisionRing K] : Field { x // x ∈ Subring.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 : { x // x ∈ Subring.center K }) : ↑a⁻¹ = (↑a)⁻¹

@[simp]

theorem Subring.center.coe_div {K : Type u} [DivisionRing K] (a : { x // x ∈ Subring.center K }) (b : { x // x ∈ Subring.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

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

@[simp]

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

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

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

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 | 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 : Set R⦄ ⦃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_induction' {R : Type u} [Ring R] {s : Set R} {p : (x : R) → x ∈ Subring.closure s → Prop} (Hs : (x : R) → (h : x ∈ s) → p x (_ : x ∈ ↑(Subring.closure s))) (H0 : p 0 (_ : 0 ∈ Subring.closure s)) (H1 : p 1 (_ : 1 ∈ Subring.closure s)) (Hadd : (x : R) → (hx : x ∈ Subring.closure s) → (y : R) → (hy : y ∈ Subring.closure s) → p x hx → p y hy → p (x + y) (_ : x + y ∈ Subring.closure s)) (Hneg : (x : R) → (hx : x ∈ Subring.closure s) → p x hx → p (-x) (_ : -x ∈ Subring.closure s)) (Hmul : (x : R) → (hx : x ∈ Subring.closure s) → (y : R) → (hy : y ∈ Subring.closure s) → p x hx → p y hy → p (x * y) (_ : x * y ∈ Subring.closure s)) {a : R} (ha : a ∈ Subring.closure s) : p a ha

theorem Subring.closure_induction₂ {R : Type u} [Ring R] {s : Set R} {p : R → R → Prop} {a : R} {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 ∈ AddSubgroup.closure ↑(Submonoid.closure s)

def Subring.closureCommRingOfComm {R : Type u} [Ring R] {s : Set R} (hcomm : ∀ (a : R), a ∈ s → ∀ (b : R), b ∈ s → a * b = b * a) : CommRing { x // x ∈ Subring.closure s }

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

Equations

• Subring.closureCommRingOfComm hcomm = let src := Subring.toRing (Subring.closure s); CommRing.mk (_ : ∀ (x y : { x // x ∈ Subring.closure s }), x * y = y * x)

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

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

closure forms a Galois insertion with the coercion to set.

Equations

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

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 : Set R) (t : Set R) : Subring.closure (s ∪ t) = Subring.closure s ⊔ Subring.closure t

theorem Subring.closure_iUnion {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) (_ : t ∈ s), Subring.closure t

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

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

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

theorem Subring.coe_prod {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring R) (t : Subring S) : ↑(Subring.prod s 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 ∈ Subring.prod s t ↔ p.fst ∈ s ∧ p.snd ∈ t

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

theorem Subring.prod_mono_right {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring R) : Monotone fun t => Subring.prod s t

theorem Subring.prod_mono_left {R : Type u} {S : Type v} [Ring R] [Ring S] (t : Subring S) : Monotone fun s => Subring.prod s t

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

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

@[simp]

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

def Subring.prodEquiv {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring R) (t : Subring S) : { x // x ∈ Subring.prod s t } ≃+* { x // x ∈ s } × { x // x ∈ t }

Product of subrings is isomorphic to their product as rings.

Equations

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

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 x x_1 => x ≤ x_1) S) {x : R} : x ∈ ⨆ (i : ι), S i ↔ ∃ i, 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 x x_1 => x ≤ x_1) 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 : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) {x : R} : x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s

theorem Subring.coe_sSup_of_directedOn {R : Type u} [Ring R] {S : Set (Subring R)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) : ↑(sSup S) = ⋃ (s : Subring R) (_ : 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 ↔ ↑(RingEquiv.symm f) 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 (↑(RingEquiv.symm f)) 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 (↑(RingEquiv.symm f)) K

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

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

This is the bundled version of Set.rangeFactorization.

Equations

• RingHom.rangeRestrict f = RingHom.codRestrict f (RingHom.range f) (_ : ∀ (x : R), ∃ y, ↑f y = ↑f x)

@[simp]

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

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

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

@[simp]

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

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

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

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

@[simp]

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

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

theorem RingHom.eq_of_eqOn_set_dense {R : Type u} {S : Type v} [Ring R] [Ring S] {s : Set R} (hs : Subring.closure s = ⊤) {f : R →+* S} {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 : Subring R} {T : Subring R} (h : S ≤ T) : { x // x ∈ S } →+* { x // x ∈ T }

The ring homomorphism associated to an inclusion of subrings.

Equations

• Subring.inclusion h = RingHom.codRestrict (Subring.subtype S) T (_ : ∀ (x : { x // x ∈ S }), ↑(Subring.subtype S) x ∈ T)

@[simp]

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

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

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

@[simp]

theorem Subring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [Ring R] [Ring S] (s : Subring R) (t : Subring S) : Subring.prod s ⊥ ⊔ Subring.prod ⊥ t = Subring.prod s t

def RingEquiv.subringCongr {R : Type u} [Ring R] {s : Subring R} {t : Subring R} (h : s = t) : { x // x ∈ s } ≃+* { x // x ∈ t }

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

Equations

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

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 ≃+* { x // x ∈ RingHom.range f }

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

Equations

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

@[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) : ↑(↑(RingEquiv.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 : { x // x ∈ RingHom.range f }) : ↑(RingEquiv.symm (RingEquiv.ofLeftInverse h)) x = g ↑x

@[simp]

theorem RingEquiv.subringMap_symm_apply_coe {R : Type u} {S : Type v} [Ring R] [Ring S] {s : Subring R} (e : R ≃+* S) : ∀ (a : ↑(↑↑(RingEquiv.toAddEquiv e) '' ↑(Subsemiring.toAddSubmonoid s.toSubsemiring))), ↑(↑(RingEquiv.symm (RingEquiv.subringMap e)) a) = ↑(AddEquiv.symm ↑e) ↑a

@[simp]

theorem RingEquiv.subringMap_apply_coe {R : Type u} {S : Type v} [Ring R] [Ring S] {s : Subring R} (e : R ≃+* S) : ∀ (a : ↑↑(Subsemiring.toAddSubmonoid s.toSubsemiring)), ↑(↑(RingEquiv.subringMap e) a) = ↑e ↑a

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

• RingEquiv.subringMap e = RingEquiv.subsemiringMap e s.toSubsemiring

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

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.instSMulSubtypeMemSubringInstMembershipInstSetLikeSubring {R : Type u} [Ring R] {α : Type u_1} [SMul R α] (S : Subring R) : SMul { x // x ∈ S } α

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

Equations

• Subring.instSMulSubtypeMemSubringInstMembershipInstSetLikeSubring S = inferInstanceAs (SMul { x // x ∈ S.toSubsemiring } α)

theorem Subring.smul_def {R : Type u} [Ring R] {α : Type u_1} [SMul R α] {S : Subring R} (g : { x // x ∈ 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 { x // x ∈ S } α β

Equations

• @Subring.smulCommClass_left = @Subring.smulCommClass_left.proof_1

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 { x // x ∈ S } β

Equations

• @Subring.smulCommClass_right = @Subring.smulCommClass_right.proof_1

instance Subring.instIsScalarTowerSubtypeMemSubringInstMembershipInstSetLikeSubringInstSMulSubtypeMemSubringInstMembershipInstSetLikeSubringInstSMulSubtypeMemSubringInstMembershipInstSetLikeSubring {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 { x // x ∈ S } α β

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

Equations

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

instance Subring.instFaithfulSMulSubtypeMemSubringInstMembershipInstSetLikeSubringInstSMulSubtypeMemSubringInstMembershipInstSetLikeSubring {R : Type u} [Ring R] {α : Type u_1} [SMul R α] [FaithfulSMul R α] (S : Subring R) : FaithfulSMul { x // x ∈ S } α

Equations

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

instance Subring.instMulActionSubtypeMemSubringInstMembershipInstSetLikeSubringToMonoidToMonoidToMonoidWithZeroToSemiringToSubmonoidToNonAssocSemiringToSubsemiring {R : Type u} [Ring R] {α : Type u_1} [MulAction R α] (S : Subring R) : MulAction { x // x ∈ S } α

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

Equations

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

instance Subring.instDistribMulActionSubtypeMemSubringInstMembershipInstSetLikeSubringToMonoidToMonoidToMonoidWithZeroToSemiringToSubmonoidToNonAssocSemiringToSubsemiring {R : Type u} [Ring R] {α : Type u_1} [AddMonoid α] [DistribMulAction R α] (S : Subring R) : DistribMulAction { x // x ∈ S } α

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

Equations

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

instance Subring.instMulDistribMulActionSubtypeMemSubringInstMembershipInstSetLikeSubringToMonoidToMonoidToMonoidWithZeroToSemiringToSubmonoidToNonAssocSemiringToSubsemiring {R : Type u} [Ring R] {α : Type u_1} [Monoid α] [MulDistribMulAction R α] (S : Subring R) : MulDistribMulAction { x // x ∈ S } α

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

Equations

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

instance Subring.instSMulWithZeroSubtypeMemSubringInstMembershipInstSetLikeSubringZeroToZeroToMonoidWithZeroToSemiringToZeroMemClassToAddZeroClassToAddMonoidToAddMonoidWithOneToAddGroupWithOneToAddSubmonoidClassToNonAssocSemiringToSubsemiringClassInstSubringClassSubringInstSetLikeSubring {R : Type u} [Ring R] {α : Type u_1} [Zero α] [SMulWithZero R α] (S : Subring R) : SMulWithZero { x // x ∈ S } α

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

Equations

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

instance Subring.instMulActionWithZeroSubtypeMemSubringInstMembershipInstSetLikeSubringToMonoidWithZeroToSemiringToSemiringToSubsemiring {R : Type u} [Ring R] {α : Type u_1} [Zero α] [MulActionWithZero R α] (S : Subring R) : MulActionWithZero { x // x ∈ S } α

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

Equations

• Subring.instMulActionWithZeroSubtypeMemSubringInstMembershipInstSetLikeSubringToMonoidWithZeroToSemiringToSemiringToSubsemiring S = Subsemiring.mulActionWithZero S.toSubsemiring

instance Subring.instModuleSubtypeMemSubringInstMembershipInstSetLikeSubringToSemiringToSemiringToSubsemiring {R : Type u} [Ring R] {α : Type u_1} [AddCommMonoid α] [Module R α] (S : Subring R) : Module { x // x ∈ S } α

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

Equations

• Subring.instModuleSubtypeMemSubringInstMembershipInstSetLikeSubringToSemiringToSemiringToSubsemiring S = Subsemiring.module S.toSubsemiring

instance Subring.instMulSemiringActionSubtypeMemSubringInstMembershipInstSetLikeSubringToMonoidToMonoidToMonoidWithZeroToSemiringToSubmonoidToNonAssocSemiringToSubsemiring {R : Type u} [Ring R] {α : Type u_1} [Semiring α] [MulSemiringAction R α] (S : Subring R) : MulSemiringAction { x // x ∈ S } α

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

Equations

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

instance Subring.center.smulCommClass_left {R : Type u} [Ring R] : SMulCommClass { x // x ∈ Subring.center R } R R

The center of a semiring acts commutatively on that semiring.

Equations

• @Subring.center.smulCommClass_left = @Subring.center.smulCommClass_left.proof_1

instance Subring.center.smulCommClass_right {R : Type u} [Ring R] : SMulCommClass R { x // x ∈ Subring.center R } R

The center of a semiring acts commutatively on that semiring.

Equations

• @Subring.center.smulCommClass_right = @Subring.center.smulCommClass_right.proof_1

def Units.posSubgroup (R : Type u_1) [LinearOrderedSemiring R] : Subgroup Rˣ

The subgroup of positive units of a linear ordered semiring.

Equations

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

@[simp]

theorem Units.mem_posSubgroup {R : Type u_1} [LinearOrderedSemiring R] (u : Rˣ) : u ∈ Units.posSubgroup R ↔ 0 < ↑u