NonUnitalSubrings

Let R be a non-unital ring. This file defines the "bundled" non-unital subring type NonUnitalSubring R, a type whose terms correspond to non-unital subrings of R. This is the preferred way to talk about non-unital subrings in mathlib.

We prove that non-unital 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 NonUnitalSubring R, sending a subset of R to the non-unital subring it generates, and prove that it is a Galois insertion.

Main definitions

Notation used here:

(R : Type u) [NonUnitalRing R] (S : Type u) [NonUnitalRing S] (f g : R →ₙ+* S) (A : NonUnitalSubring R) (B : NonUnitalSubring S) (s : Set R)

• NonUnitalSubring R : the type of non-unital subrings of a ring R.

• instance : CompleteLattice (NonUnitalSubring R) : the complete lattice structure on the non-unital subrings.

• NonUnitalSubring.center : the center of a non-unital ring R.

• NonUnitalSubring.closure : non-unital subring closure of a set, i.e., the smallest non-unital subring that includes the set.

• NonUnitalSubring.gi : closure : Set M → NonUnitalSubring M and coercion coe : NonUnitalSubring M → Set M form a GaloisInsertion.

• comap f B : NonUnitalSubring A : the preimage of a non-unital subring B along the non-unital ring homomorphism f

• map f A : NonUnitalSubring B : the image of a non-unital subring A along the non-unital ring homomorphism f.

• Prod A B : NonUnitalSubring (R × S) : the product of non-unital subrings

• f.range : NonUnitalSubring B : the range of the non-unital ring homomorphism f.

• eq_locus f g : NonUnitalSubring R : given non-unital ring homomorphisms f g : R →ₙ+* S, the non-unital subring of R where f x = g x

Implementation notes

A non-unital subring is implemented as a NonUnitalSubsemiring 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 non-unital subring's underlying set.

Tags

non-unital subring

class NonUnitalSubringClass (S : Type u_1) (R : Type u) [NonUnitalNonAssocRing R] [SetLike S R] extends NonUnitalSubsemiringClass , NegMemClass : Prop

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

@[instance 100]

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

Equations

• ⋯ = ⋯

@[instance 75]

instance NonUnitalSubringClass.toNonUnitalNonAssocRing {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S) : NonUnitalNonAssocRing ↥s

A non-unital subring of a non-unital ring inherits a non-unital ring structure

Equations

• NonUnitalSubringClass.toNonUnitalNonAssocRing s = Function.Injective.nonUnitalNonAssocRing Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[instance 75]

instance NonUnitalSubringClass.toNonUnitalRing {S : Type v} {R : Type u_1} [NonUnitalRing R] [SetLike S R] [NonUnitalSubringClass S R] (s : S) : NonUnitalRing ↥s

A non-unital subring of a non-unital ring inherits a non-unital ring structure

Equations

• NonUnitalSubringClass.toNonUnitalRing s = Function.Injective.nonUnitalRing Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[instance 75]

instance NonUnitalSubringClass.toNonUnitalCommRing {S : Type v} (s : S) {R : Type u_1} [NonUnitalCommRing R] [SetLike S R] [NonUnitalSubringClass S R] : NonUnitalCommRing ↥s

A non-unital subring of a NonUnitalCommRing is a NonUnitalCommRing.

Equations

• NonUnitalSubringClass.toNonUnitalCommRing s = Function.Injective.nonUnitalCommRing Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

def NonUnitalSubringClass.subtype {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S) : ↥s →ₙ+* R

The natural non-unital ring hom from a non-unital subring of a non-unital ring R to R.

Equations

• NonUnitalSubringClass.subtype s = { toFun := Subtype.val, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem NonUnitalSubringClass.coe_subtype {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S) : ⇑(NonUnitalSubringClass.subtype s) = Subtype.val

structure NonUnitalSubring (R : Type u) [NonUnitalNonAssocRing R] extends NonUnitalSubsemiring : Type u

NonUnitalSubring R is the type of non-unital subrings of R. A non-unital subring of R is a subset s that is a multiplicative subsemigroup and an additive subgroup. Note in particular that it shares the same 0 as R.

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

  G is closed under negation

@[reducible]

abbrev NonUnitalSubring.toAddSubgroup {R : Type u} [NonUnitalNonAssocRing R] (self : NonUnitalSubring R) : AddSubgroup R

Reinterpret a NonUnitalSubring as an AddSubgroup.

Equations

• self.toAddSubgroup = { toAddSubmonoid := self.toAddSubmonoid, neg_mem' := ⋯ }

def NonUnitalSubring.toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : Subsemigroup R

The underlying submonoid of a NonUnitalSubring.

Equations

• s.toSubsemigroup = { carrier := s.carrier, mul_mem' := ⋯ }

instance NonUnitalSubring.instSetLike {R : Type u} [NonUnitalNonAssocRing R] : SetLike (NonUnitalSubring R) R

Equations

• NonUnitalSubring.instSetLike = { coe := fun (s : NonUnitalSubring R) => s.carrier, coe_injective' := ⋯ }

instance NonUnitalSubring.instNonUnitalSubringClass {R : Type u} [NonUnitalNonAssocRing R] : NonUnitalSubringClass (NonUnitalSubring R) R

Equations

• ⋯ = ⋯

theorem NonUnitalSubring.mem_carrier {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} : x ∈ s.toNonUnitalSubsemiring ↔ x ∈ s

@[simp]

theorem NonUnitalSubring.mem_mk {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubsemiring R} {x : R} (h : ∀ {x : R}, x ∈ S.carrier → -x ∈ S.carrier) : x ∈ { toNonUnitalSubsemiring := S, neg_mem' := h } ↔ x ∈ S

@[simp]

theorem NonUnitalSubring.coe_set_mk {R : Type u} [NonUnitalNonAssocRing R] (S : NonUnitalSubsemiring R) (h : ∀ {x : R}, x ∈ S.carrier → -x ∈ S.carrier) : ↑{ toNonUnitalSubsemiring := S, neg_mem' := h } = ↑S

@[simp]

theorem NonUnitalSubring.mk_le_mk {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubsemiring R} {S' : NonUnitalSubsemiring R} (h : ∀ {x : R}, x ∈ S.carrier → -x ∈ S.carrier) (h' : ∀ {x : R}, x ∈ S'.carrier → -x ∈ S'.carrier) : { toNonUnitalSubsemiring := S, neg_mem' := h } ≤ { toNonUnitalSubsemiring := S', neg_mem' := h' } ↔ S ≤ S'

theorem NonUnitalSubring.ext_iff {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubring R} {T : NonUnitalSubring R} : S = T ↔ ∀ (x : R), x ∈ S ↔ x ∈ T

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

Two non-unital subrings are equal if they have the same elements.

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

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

Equations

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

@[simp]

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

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

theorem NonUnitalSubring.toNonUnitalSubsemiring_injective {R : Type u} [NonUnitalNonAssocRing R] : Function.Injective NonUnitalSubring.toNonUnitalSubsemiring

theorem NonUnitalSubring.toNonUnitalSubsemiring_strictMono {R : Type u} [NonUnitalNonAssocRing R] : StrictMono NonUnitalSubring.toNonUnitalSubsemiring

theorem NonUnitalSubring.toNonUnitalSubsemiring_mono {R : Type u} [NonUnitalNonAssocRing R] : Monotone NonUnitalSubring.toNonUnitalSubsemiring

theorem NonUnitalSubring.toAddSubgroup_injective {R : Type u} [NonUnitalNonAssocRing R] : Function.Injective NonUnitalSubring.toAddSubgroup

theorem NonUnitalSubring.toAddSubgroup_strictMono {R : Type u} [NonUnitalNonAssocRing R] : StrictMono NonUnitalSubring.toAddSubgroup

theorem NonUnitalSubring.toAddSubgroup_mono {R : Type u} [NonUnitalNonAssocRing R] : Monotone NonUnitalSubring.toAddSubgroup

theorem NonUnitalSubring.toSubsemigroup_injective {R : Type u} [NonUnitalNonAssocRing R] : Function.Injective NonUnitalSubring.toSubsemigroup

theorem NonUnitalSubring.toSubsemigroup_strictMono {R : Type u} [NonUnitalNonAssocRing R] : StrictMono NonUnitalSubring.toSubsemigroup

theorem NonUnitalSubring.toSubsemigroup_mono {R : Type u} [NonUnitalNonAssocRing R] : Monotone NonUnitalSubring.toSubsemigroup

def NonUnitalSubring.mk' {R : Type u} [NonUnitalNonAssocRing R] (s : Set R) (sm : Subsemigroup R) (sa : AddSubgroup R) (hm : ↑sm = s) (ha : ↑sa = s) : NonUnitalSubring R

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

Equations

• NonUnitalSubring.mk' s sm sa hm ha = { carrier := (sm.copy s ⋯).carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯, neg_mem' := ⋯ }

@[simp]

theorem NonUnitalSubring.coe_mk' {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R} (ha : ↑sa = s) : ↑(NonUnitalSubring.mk' s sm sa hm ha) = s

@[simp]

theorem NonUnitalSubring.mem_mk' {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R} (ha : ↑sa = s) {x : R} : x ∈ NonUnitalSubring.mk' s sm sa hm ha ↔ x ∈ s

@[simp]

theorem NonUnitalSubring.mk'_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R} (ha : ↑sa = s) : (NonUnitalSubring.mk' s sm sa hm ha).toSubsemigroup = sm

@[simp]

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

theorem NonUnitalSubring.zero_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : 0 ∈ s

A non-unital subring contains the ring's 0.

theorem NonUnitalSubring.mul_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} {y : R} : x ∈ s → y ∈ s → x * y ∈ s

A non-unital subring is closed under multiplication.

theorem NonUnitalSubring.add_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} {y : R} : x ∈ s → y ∈ s → x + y ∈ s

A non-unital subring is closed under addition.

theorem NonUnitalSubring.neg_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} : x ∈ s → -x ∈ s

A non-unital subring is closed under negation.

theorem NonUnitalSubring.sub_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} {y : R} (hx : x ∈ s) (hy : y ∈ s) : x - y ∈ s

A non-unital subring is closed under subtraction

theorem NonUnitalSubring.list_sum_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {l : List R} : (∀ x ∈ l, x ∈ s) → l.sum ∈ s

Sum of a list of elements in a non-unital subring is in the non-unital subring.

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

Sum of a multiset of elements in a NonUnitalSubring of a NonUnitalRing is in the NonUnitalSubring.

theorem NonUnitalSubring.sum_mem {R : Type u_1} [NonUnitalNonAssocRing R] (s : NonUnitalSubring 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 NonUnitalSubring of a NonUnitalRing indexed by a Finset is in the NonUnitalSubring.

instance NonUnitalSubring.toNonUnitalRing {R : Type u_1} [NonUnitalRing R] (s : NonUnitalSubring R) : NonUnitalRing ↥s

A non-unital subring of a non-unital ring inherits a non-unital ring structure

Equations

• s.toNonUnitalRing = Function.Injective.nonUnitalRing (fun (a : ↥s) => ↑a) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

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

@[simp]

theorem NonUnitalSubring.val_add {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (x : ↥s) (y : ↥s) : ↑(x + y) = ↑x + ↑y

@[simp]

theorem NonUnitalSubring.val_neg {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (x : ↥s) : ↑(-x) = -↑x

@[simp]

theorem NonUnitalSubring.val_mul {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (x : ↥s) (y : ↥s) : ↑(x * y) = ↑x * ↑y

@[simp]

theorem NonUnitalSubring.val_zero {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : ↑0 = 0

theorem NonUnitalSubring.coe_eq_zero_iff {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : ↥s} : ↑x = 0 ↔ x = 0

instance NonUnitalSubring.toNonUnitalCommRing {R : Type u_1} [NonUnitalCommRing R] (s : NonUnitalSubring R) : NonUnitalCommRing ↥s

A non-unital subring of a NonUnitalCommRing is a NonUnitalCommRing.

Equations

• s.toNonUnitalCommRing = Function.Injective.nonUnitalCommRing (fun (a : ↥s) => ↑a) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

Partial order

@[simp]

theorem NonUnitalSubring.mem_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} : x ∈ s.toSubsemigroup ↔ x ∈ s

@[simp]

theorem NonUnitalSubring.coe_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : ↑s.toSubsemigroup = ↑s

@[simp]

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

@[simp]

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

@[simp]

theorem NonUnitalSubring.mem_toNonUnitalSubsemiring {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} : x ∈ s.toNonUnitalSubsemiring ↔ x ∈ s

@[simp]

theorem NonUnitalSubring.coe_toNonUnitalSubsemiring {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : ↑s.toNonUnitalSubsemiring = ↑s

top

instance NonUnitalSubring.instTop {R : Type u} [NonUnitalNonAssocRing R] : Top (NonUnitalSubring R)

The non-unital subring R of the ring R.

Equations

• NonUnitalSubring.instTop = { top := let __src := ⊤; let __src_1 := ⊤; { carrier := __src.carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯, neg_mem' := ⋯ } }

@[simp]

theorem NonUnitalSubring.mem_top {R : Type u} [NonUnitalNonAssocRing R] (x : R) : x ∈ ⊤

@[simp]

theorem NonUnitalSubring.coe_top {R : Type u} [NonUnitalNonAssocRing R] : ↑⊤ = Set.univ

@[simp]

theorem NonUnitalSubring.topEquiv_apply {R : Type u} [NonUnitalNonAssocRing R] (x : ↥⊤) : NonUnitalSubring.topEquiv x = ↑x

@[simp]

theorem NonUnitalSubring.topEquiv_symm_apply_coe {R : Type u} [NonUnitalNonAssocRing R] (x : R) : ↑(NonUnitalSubring.topEquiv.symm x) = x

def NonUnitalSubring.topEquiv {R : Type u} [NonUnitalNonAssocRing R] : ↥⊤ ≃+* R

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

Equations

• NonUnitalSubring.topEquiv = NonUnitalSubsemiring.topEquiv

comap

def NonUnitalSubring.comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring S) : NonUnitalSubring R

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

Equations

• NonUnitalSubring.comap f s = { carrier := ⇑f ⁻¹' s.carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯, neg_mem' := ⋯ }

@[simp]

theorem NonUnitalSubring.coe_comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring S) (f : F) : ↑(NonUnitalSubring.comap f s) = ⇑f ⁻¹' ↑s

@[simp]

theorem NonUnitalSubring.mem_comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {s : NonUnitalSubring S} {f : F} {x : R} : x ∈ NonUnitalSubring.comap f s ↔ f x ∈ s

theorem NonUnitalSubring.comap_comap {R : Type u} {S : Type v} {T : Type u_1} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T] (s : NonUnitalSubring T) (g : S →ₙ+* T) (f : R →ₙ+* S) : NonUnitalSubring.comap f (NonUnitalSubring.comap g s) = NonUnitalSubring.comap (g.comp f) s

map

def NonUnitalSubring.map {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring R) : NonUnitalSubring S

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

Equations

• NonUnitalSubring.map f s = { carrier := ⇑f '' s.carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯, neg_mem' := ⋯ }

@[simp]

theorem NonUnitalSubring.coe_map {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring R) : ↑(NonUnitalSubring.map f s) = ⇑f '' ↑s

@[simp]

theorem NonUnitalSubring.mem_map {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {s : NonUnitalSubring R} {y : S} : y ∈ NonUnitalSubring.map f s ↔ ∃ x ∈ s, f x = y

@[simp]

theorem NonUnitalSubring.map_id {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : NonUnitalSubring.map (NonUnitalRingHom.id R) s = s

theorem NonUnitalSubring.map_map {R : Type u} {S : Type v} {T : Type u_1} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T] (s : NonUnitalSubring R) (g : S →ₙ+* T) (f : R →ₙ+* S) : NonUnitalSubring.map g (NonUnitalSubring.map f s) = NonUnitalSubring.map (g.comp f) s

theorem NonUnitalSubring.map_le_iff_le_comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {s : NonUnitalSubring R} {t : NonUnitalSubring S} : NonUnitalSubring.map f s ≤ t ↔ s ≤ NonUnitalSubring.comap f t

theorem NonUnitalSubring.gc_map_comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : GaloisConnection (NonUnitalSubring.map f) (NonUnitalSubring.comap f)

noncomputable def NonUnitalSubring.equivMapOfInjective {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R) (f : F) (hf : Function.Injective ⇑f) : ↥s ≃+* ↥(NonUnitalSubring.map f s)

A NonUnitalSubring 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 NonUnitalSubring.coe_equivMapOfInjective_apply {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R) (f : F) (hf : Function.Injective ⇑f) (x : ↥s) : ↑((s.equivMapOfInjective f hf) x) = f ↑x

range

def NonUnitalRingHom.range {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) : NonUnitalSubring S

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

Equations

• f.range = (NonUnitalSubring.map f ⊤).copy (Set.range ⇑f) ⋯

@[simp]

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

@[simp]

theorem NonUnitalRingHom.mem_range {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R →ₙ+* S} {y : S} : y ∈ f.range ↔ ∃ (x : R), f x = y

theorem NonUnitalRingHom.range_eq_map {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) : f.range = NonUnitalSubring.map f ⊤

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

theorem NonUnitalRingHom.map_range {R : Type u} {S : Type v} {T : Type u_1} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T] (g : S →ₙ+* T) (f : R →ₙ+* S) : NonUnitalSubring.map g f.range = (g.comp f).range

instance NonUnitalRingHom.fintypeRange {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [Fintype R] [DecidableEq S] (f : R →ₙ+* 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 NonUnitalSubring.instBot {R : Type u} [NonUnitalNonAssocRing R] : Bot (NonUnitalSubring R)

Equations

• NonUnitalSubring.instBot = { bot := NonUnitalRingHom.range 0 }

instance NonUnitalSubring.instInhabited {R : Type u} [NonUnitalNonAssocRing R] : Inhabited (NonUnitalSubring R)

Equations

• NonUnitalSubring.instInhabited = { default := ⊥ }

theorem NonUnitalSubring.coe_bot {R : Type u} [NonUnitalNonAssocRing R] : ↑⊥ = {0}

theorem NonUnitalSubring.mem_bot {R : Type u} [NonUnitalNonAssocRing R] {x : R} : x ∈ ⊥ ↔ x = 0

inf

instance NonUnitalSubring.instInf {R : Type u} [NonUnitalNonAssocRing R] : Inf (NonUnitalSubring R)

The inf of two NonUnitalSubrings is their intersection.

Equations

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

@[simp]

theorem NonUnitalSubring.coe_inf {R : Type u} [NonUnitalNonAssocRing R] (p : NonUnitalSubring R) (p' : NonUnitalSubring R) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

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

instance NonUnitalSubring.instInfSet {R : Type u} [NonUnitalNonAssocRing R] : InfSet (NonUnitalSubring R)

Equations

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

@[simp]

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

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

@[simp]

theorem NonUnitalSubring.coe_iInf {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_2} {S : ι → NonUnitalSubring R} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

theorem NonUnitalSubring.mem_iInf {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_2} {S : ι → NonUnitalSubring R} {x : R} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

@[simp]

theorem NonUnitalSubring.sInf_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] (s : Set (NonUnitalSubring R)) : (sInf s).toSubsemigroup = ⨅ t ∈ s, t.toSubsemigroup

@[simp]

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

instance NonUnitalSubring.instCompleteLattice {R : Type u} [NonUnitalNonAssocRing R] : CompleteLattice (NonUnitalSubring R)

NonUnitalSubrings of a ring form a complete lattice.

Equations

• NonUnitalSubring.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

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

Center of a ring

def NonUnitalSubring.center (R : Type u) [NonUnitalNonAssocRing R] : NonUnitalSubring R

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

Equations

• NonUnitalSubring.center R = { toNonUnitalSubsemiring := NonUnitalSubsemiring.center R, neg_mem' := ⋯ }

theorem NonUnitalSubring.coe_center (R : Type u) [NonUnitalNonAssocRing R] : ↑(NonUnitalSubring.center R) = Set.center R

@[simp]

theorem NonUnitalSubring.center_toNonUnitalSubsemiring (R : Type u) [NonUnitalNonAssocRing R] : (NonUnitalSubring.center R).toNonUnitalSubsemiring = NonUnitalSubsemiring.center R

instance NonUnitalSubring.center.instNonUnitalCommRing (R : Type u) [NonUnitalNonAssocRing R] : NonUnitalCommRing ↥(NonUnitalSubring.center R)

The center is commutative and associative.

Equations

• NonUnitalSubring.center.instNonUnitalCommRing R = NonUnitalCommRing.mk ⋯

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

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

Equations

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

@[simp]

theorem NonUnitalSubring.center_eq_top (R : Type u_1) [NonUnitalCommRing R] : NonUnitalSubring.center R = ⊤

NonUnitalSubring closure of a subset

def NonUnitalSubring.closure {R : Type u} [NonUnitalNonAssocRing R] (s : Set R) : NonUnitalSubring R

The NonUnitalSubring generated by a set.

Equations

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

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

@[simp]

theorem NonUnitalSubring.subset_closure {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} : s ⊆ ↑(NonUnitalSubring.closure s)

The NonUnitalSubring generated by a set includes the set.

theorem NonUnitalSubring.not_mem_of_not_mem_closure {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {P : R} (hP : P ∉ NonUnitalSubring.closure s) : P ∉ s

@[simp]

theorem NonUnitalSubring.closure_le {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {t : NonUnitalSubring R} : NonUnitalSubring.closure s ≤ t ↔ s ⊆ ↑t

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

theorem NonUnitalSubring.closure_mono {R : Type u} [NonUnitalNonAssocRing R] ⦃s : Set R⦄ ⦃t : Set R⦄ (h : s ⊆ t) : NonUnitalSubring.closure s ≤ NonUnitalSubring.closure t

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

theorem NonUnitalSubring.closure_eq_of_le {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {t : NonUnitalSubring R} (h₁ : s ⊆ ↑t) (h₂ : t ≤ NonUnitalSubring.closure s) : NonUnitalSubring.closure s = t

theorem NonUnitalSubring.closure_induction {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {p : R → Prop} {x : R} (h : x ∈ NonUnitalSubring.closure s) (mem : ∀ x ∈ s, p x) (zero : p 0) (add : ∀ (x y : R), p x → p y → p (x + y)) (neg : ∀ (x : R), p x → p (-x)) (mul : ∀ (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 NonUnitalSubring.closure_induction' {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {p : ↥(NonUnitalSubring.closure s) → Prop} (a : ↥(NonUnitalSubring.closure s)) (mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩) (zero : p 0) (add : ∀ (x y : ↥(NonUnitalSubring.closure s)), p x → p y → p (x + y)) (neg : ∀ (x : ↥(NonUnitalSubring.closure s)), p x → p (-x)) (mul : ∀ (x y : ↥(NonUnitalSubring.closure s)), p x → p y → p (x * y)) : p a

The difference with NonUnitalSubring.closure_induction is that this acts on the subtype.

theorem NonUnitalSubring.closure_induction₂ {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {p : R → R → Prop} {a : R} {b : R} (ha : a ∈ NonUnitalSubring.closure s) (hb : b ∈ NonUnitalSubring.closure s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ (x : R), p 0 x) (H0_right : ∀ (x : R), p x 0) (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 NonUnitalSubring.mem_closure_iff {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {x : R} : x ∈ NonUnitalSubring.closure s ↔ x ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)

def NonUnitalSubring.closureNonUnitalCommRingOfComm {R : Type u} [NonUnitalRing R] {s : Set R} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : NonUnitalCommRing ↥(NonUnitalSubring.closure s)

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

Equations

• NonUnitalSubring.closureNonUnitalCommRingOfComm hcomm = NonUnitalCommRing.mk ⋯

def NonUnitalSubring.gi (R : Type u) [NonUnitalNonAssocRing R] : GaloisInsertion NonUnitalSubring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

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

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

Closure of a NonUnitalSubring S equals S.

@[simp]

theorem NonUnitalSubring.closure_empty {R : Type u} [NonUnitalNonAssocRing R] : NonUnitalSubring.closure ∅ = ⊥

@[simp]

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

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

theorem NonUnitalSubring.closure_iUnion {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_2} (s : ι → Set R) : NonUnitalSubring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), NonUnitalSubring.closure (s i)

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

theorem NonUnitalSubring.map_sup {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R) (t : NonUnitalSubring R) (f : F) : NonUnitalSubring.map f (s ⊔ t) = NonUnitalSubring.map f s ⊔ NonUnitalSubring.map f t

theorem NonUnitalSubring.map_iSup {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} (f : F) (s : ι → NonUnitalSubring R) : NonUnitalSubring.map f (iSup s) = ⨆ (i : ι), NonUnitalSubring.map f (s i)

theorem NonUnitalSubring.map_inf {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R) (t : NonUnitalSubring R) (f : F) (hf : Function.Injective ⇑f) : NonUnitalSubring.map f (s ⊓ t) = NonUnitalSubring.map f s ⊓ NonUnitalSubring.map f t

theorem NonUnitalSubring.map_iInf {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} [Nonempty ι] (f : F) (hf : Function.Injective ⇑f) (s : ι → NonUnitalSubring R) : NonUnitalSubring.map f (iInf s) = ⨅ (i : ι), NonUnitalSubring.map f (s i)

theorem NonUnitalSubring.comap_inf {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring S) (t : NonUnitalSubring S) (f : F) : NonUnitalSubring.comap f (s ⊓ t) = NonUnitalSubring.comap f s ⊓ NonUnitalSubring.comap f t

theorem NonUnitalSubring.comap_iInf {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} (f : F) (s : ι → NonUnitalSubring S) : NonUnitalSubring.comap f (iInf s) = ⨅ (i : ι), NonUnitalSubring.comap f (s i)

@[simp]

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

@[simp]

theorem NonUnitalSubring.comap_top {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) : NonUnitalSubring.comap f ⊤ = ⊤

def NonUnitalSubring.prod {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring R) (t : NonUnitalSubring S) : NonUnitalSubring (R × S)

Given NonUnitalSubrings s, t of rings R, S respectively, s.prod t is s ×ˢ t as a NonUnitalSubring of R × S.

Equations

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

theorem NonUnitalSubring.coe_prod {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring R) (t : NonUnitalSubring S) : ↑(s.prod t) = ↑s ×ˢ ↑t

theorem NonUnitalSubring.mem_prod {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {s : NonUnitalSubring R} {t : NonUnitalSubring S} {p : R × S} : p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t

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

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

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

theorem NonUnitalSubring.prod_top {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring R) : s.prod ⊤ = NonUnitalSubring.comap (NonUnitalRingHom.fst R S) s

theorem NonUnitalSubring.top_prod {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring S) : ⊤.prod s = NonUnitalSubring.comap (NonUnitalRingHom.snd R S) s

@[simp]

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

def NonUnitalSubring.prodEquiv {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring R) (t : NonUnitalSubring S) : ↥(s.prod t) ≃+* ↥s × ↥t

Product of NonUnitalSubrings is isomorphic to their product as rings.

Equations

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

theorem NonUnitalSubring.mem_iSup_of_directed {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_2} [hι : Nonempty ι] {S : ι → NonUnitalSubring R} (hS : Directed (fun (x1 x2 : NonUnitalSubring R) => x1 ≤ x2) S) {x : R} : x ∈ ⨆ (i : ι), S i ↔ ∃ (i : ι), x ∈ S i

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

theorem NonUnitalSubring.coe_iSup_of_directed {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_2} [Nonempty ι] {S : ι → NonUnitalSubring R} (hS : Directed (fun (x1 x2 : NonUnitalSubring R) => x1 ≤ x2) S) : ↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

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

theorem NonUnitalSubring.coe_sSup_of_directedOn {R : Type u} [NonUnitalNonAssocRing R] {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : NonUnitalSubring R) => x1 ≤ x2) S) : ↑(sSup S) = ⋃ s ∈ S, ↑s

theorem NonUnitalSubring.mem_map_equiv {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R ≃+* S} {K : NonUnitalSubring R} {x : S} : x ∈ NonUnitalSubring.map (↑f) K ↔ f.symm x ∈ K

theorem NonUnitalSubring.map_equiv_eq_comap_symm {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R ≃+* S) (K : NonUnitalSubring R) : NonUnitalSubring.map (↑f) K = NonUnitalSubring.comap f.symm K

theorem NonUnitalSubring.comap_equiv_eq_map_symm {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R ≃+* S) (K : NonUnitalSubring S) : NonUnitalSubring.comap (↑f) K = NonUnitalSubring.map f.symm K

def NonUnitalRingHom.rangeRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) : R →ₙ+* ↥f.range

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

This is the bundled version of Set.rangeFactorization.

Equations

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

@[simp]

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

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

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

@[simp]

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

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

def NonUnitalRingHom.eqLocus {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) (g : R →ₙ+* S) : NonUnitalSubring R

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

Equations

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

@[simp]

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

theorem NonUnitalRingHom.eqOn_set_closure {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R →ₙ+* S} {g : R →ₙ+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(NonUnitalSubring.closure s)

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

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

theorem NonUnitalRingHom.eq_of_eqOn_set_dense {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {s : Set R} (hs : NonUnitalSubring.closure s = ⊤) {f : R →ₙ+* S} {g : R →ₙ+* S} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem NonUnitalRingHom.closure_preimage_le {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) (s : Set S) : NonUnitalSubring.closure (⇑f ⁻¹' s) ≤ NonUnitalSubring.comap f (NonUnitalSubring.closure s)

theorem NonUnitalRingHom.map_closure {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) (s : Set R) : NonUnitalSubring.map f (NonUnitalSubring.closure s) = NonUnitalSubring.closure (⇑f '' s)

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

def NonUnitalSubring.inclusion {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubring R} {T : NonUnitalSubring R} (h : S ≤ T) : ↥S →ₙ+* ↥T

The ring homomorphism associated to an inclusion of NonUnitalSubrings.

Equations

• NonUnitalSubring.inclusion h = NonUnitalRingHom.codRestrict (NonUnitalSubringClass.subtype S) T ⋯

@[simp]

theorem NonUnitalSubring.range_subtype {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) : (NonUnitalSubringClass.subtype s).range = s

theorem NonUnitalSubring.range_fst {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] : NonUnitalRingHom.srange (NonUnitalRingHom.fst R S) = ⊤

theorem NonUnitalSubring.range_snd {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] : NonUnitalRingHom.srange (NonUnitalRingHom.snd R S) = ⊤

def RingEquiv.nonUnitalSubringCongr {R : Type u} [NonUnitalRing R] {s : NonUnitalSubring R} {t : NonUnitalSubring R} (h : s = t) : ↥s ≃+* ↥t

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

Equations

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

def RingEquiv.ofLeftInverse' {R : Type u} {S : Type v} [NonUnitalRing R] [NonUnitalRing S] {g : S → R} {f : R →ₙ+* 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

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

@[simp]

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

theorem NonUnitalSubring.closure_preimage_le {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : Set S) : NonUnitalSubring.closure (⇑f ⁻¹' s) ≤ NonUnitalSubring.comap f (NonUnitalSubring.closure s)