# Documentation

Mathlib.Deprecated.Subring

# 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 RingTheory.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 , :

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

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

Construct a Subring from a set satisfying IsSubring.

Instances For
theorem RingHom.isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S} (hs : ) :
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 (f '' s)
theorem RingHom.isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
theorem IsSubring.inter {R : Type u} [Ring R] {S₁ : Set R} {S₂ : Set R} (hS₁ : ) (hS₂ : ) :
IsSubring (S₁ S₂)
theorem IsSubring.iInter {R : Type u} [Ring R] {ι : Sort u_1} {S : ιSet R} (h : ∀ (y : ι), IsSubring (S y)) :
theorem isSubring_iUnion_of_directed {R : Type u} [Ring R] {ι : Type u_1} [] {s : ιSet R} (h : ∀ (i : ι), IsSubring (s i)) (directed : ∀ (i j : ι), k, 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.

Instances For
theorem Ring.exists_list_of_mem_closure {R : Type u} [Ring R] {s : Set R} {a : R} (h : ) :
L, (∀ (l : List R), l L∀ (x : R), x lx s x = -1) List.sum (List.map List.prod L) = a
theorem Ring.InClosure.recOn {R : Type u} [Ring R] {s : Set R} {C : RProp} {x : R} (hx : ) (h1 : C 1) (hneg1 : C (-1)) (hs : (z : R) → z s(n : R) → C nC (z * n)) (ha : {x y : R} → C xC yC (x + y)) :
C x
theorem Ring.closure.isSubring {R : Type u} [Ring R] {s : Set R} :
theorem Ring.mem_closure {R : Type u} [Ring R] {s : Set R} {a : R} :
a s
theorem Ring.subset_closure {R : Type u} [Ring R] {s : Set R} :
theorem Ring.closure_subset {R : Type u} [Ring R] {s : Set R} {t : Set R} (ht : ) :
s t
theorem Ring.closure_subset_iff {R : Type u} [Ring R] {s : Set R} {t : Set R} (ht : ) :
s t
theorem Ring.closure_mono {R : Type u} [Ring R] {s : Set R} {t : Set R} (H : s 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 (f '' s)