Unbundled subrings (deprecated)

This file is deprecated, and is no longer imported by anything in mathlib other than other deprecated files, and test files. You should not need to import it.

This file defines predicates for unbundled subrings. Instead of using this file, please use Subring, defined in Mathlib.Algebra.Ring.Subring.Basic, for subrings of rings.

Main definitions

IsSubring (S : Set R) : Prop : the predicate that S is the underlying set of a subring of the ring R. The bundled variant Subring R should be used in preference to this.

Tags

IsSubring

structure IsSubring {R : Type u} [Ring R] (S : Set R) extends IsAddSubgroup S, IsSubmonoid S : Prop

S is a subring: a set containing 1 and closed under multiplication, addition and additive inverse.

• zero_mem : 0 ∈ S
• add_mem : ∀ {a b : R}, a ∈ S → b ∈ S → a + b ∈ S
• neg_mem : ∀ {a : R}, a ∈ S → -a ∈ S
• one_mem : 1 ∈ S
• mul_mem : ∀ {a b : R}, a ∈ S → b ∈ S → a * b ∈ S

def IsSubring.subring {R : Type u} [Ring R] {S : Set R} (hs : IsSubring S) : Subring R

Construct a Subring from a set satisfying IsSubring.

Equations

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

theorem RingHom.isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S} (hs : IsSubring s) : IsSubring (⇑f ⁻¹' s)

theorem RingHom.isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R} (hs : IsSubring s) : IsSubring (⇑f '' s)

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

theorem IsSubring.inter {R : Type u} [Ring R] {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : IsSubring S₂) : IsSubring (S₁ ∩ S₂)

theorem IsSubring.iInter {R : Type u} [Ring R] {ι : Sort u_1} {S : ι → Set R} (h : ∀ (y : ι), IsSubring (S y)) : IsSubring (Set.iInter S)

theorem isSubring_iUnion_of_directed {R : Type u} [Ring R] {ι : Type u_1} [Nonempty ι] {s : ι → Set R} (h : ∀ (i : ι), IsSubring (s i)) (directed : ∀ (i j : ι), ∃ (k : ι), s i ⊆ s k ∧ s j ⊆ s k) : IsSubring (⋃ (i : ι), s i)

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

The smallest subring containing a given subset of a ring, considered as a set. This function is deprecated; use Subring.closure.

Equations

• Ring.closure s = AddGroup.closure (Monoid.Closure s)

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

theorem Ring.InClosure.recOn {R : Type u} [Ring R] {s : Set R} {C : R → Prop} {x : R} (hx : x ∈ Ring.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 Ring.closure.isSubring {R : Type u} [Ring R] {s : Set R} : IsSubring (Ring.closure s)

theorem Ring.mem_closure {R : Type u} [Ring R] {s : Set R} {a : R} : a ∈ s → a ∈ Ring.closure s

theorem Ring.subset_closure {R : Type u} [Ring R] {s : Set R} : s ⊆ Ring.closure s

theorem Ring.closure_subset {R : Type u} [Ring R] {s t : Set R} (ht : IsSubring t) : s ⊆ t → Ring.closure s ⊆ t

theorem Ring.closure_subset_iff {R : Type u} [Ring R] {s t : Set R} (ht : IsSubring t) : Ring.closure s ⊆ t ↔ s ⊆ t

theorem Ring.closure_mono {R : Type u} [Ring R] {s t : Set R} (H : s ⊆ t) : Ring.closure s ⊆ Ring.closure t

theorem Ring.image_closure {R : Type u} [Ring R] {S : Type u_1} [Ring S] (f : R →+* S) (s : Set R) : ⇑f '' Ring.closure s = Ring.closure (⇑f '' s)