mathlib documentation

ring_theory.free_comm_ring

Free commutative rings

The theory of the free commutative ring generated by a type α. It is isomorphic to the polynomial ring over ℤ with variables in α

Main definitions

free_comm_ring α : the free commutative ring on a type α
lift_hom (f : α → R) : the ring hom free_comm_ring α →+* R induced by functoriality from f.
map (f : α → β) : the ring hom free_comm_ring α →*+ free_comm_ring β induced by functoriality from f.

Main results

free_comm_ring has functorial properties (it is an adjoint to the forgetful functor). In this file we have:

of : α → free_comm_ring α
lift_hom (f : α → R) : free_comm_ring α →+* R
map (f : α → β) : free_comm_ring α →+* free_comm_ring β
free_comm_ring_equiv_mv_polynomial_int : free_comm_ring α ≃+* mv_polynomial α ℤ : free_comm_ring α is isomorphic to a polynomial ring.

Implementation notes

free_comm_ring α is implemented not using mv_polynomial but directly as the free abelian group on multiset α, the type of monomials in this free commutative ring.

Tags

free commutative ring, free ring

def free_comm_ring (α : Type u) :

Type u

free_comm_ring α is the free commutative ring on the type α.

Equations

free_comm_ring α = free_abelian_group (multiplicative (multiset α))

@[instance]

def free_comm_ring.comm_ring (α : Type u) :

comm_ring (free_comm_ring α)

The structure of a commutative ring on free_comm_ring α.

Equations

free_comm_ring.comm_ring α = free_abelian_group.comm_ring (multiplicative (multiset α))

@[instance]

def free_comm_ring.inhabited (α : Type u) :

inhabited (free_comm_ring α)

Equations

free_comm_ring.inhabited α = {default := 0}

def free_comm_ring.of {α : Type u} (x : α) :

free_comm_ring α

The canonical map from α to the free commutative ring on α.

Equations

free_comm_ring.of x = free_abelian_group.of ↑[x]

theorem free_comm_ring.induction_on {α : Type u} {C : free_comm_ring α → Prop} (z : free_comm_ring α) (hn1 : C (-1)) (hb : ∀ (b : α), C (free_comm_ring.of b)) (ha : ∀ (x y : free_comm_ring α), C x → C y → C (x + y)) (hm : ∀ (x y : free_comm_ring α), C x → C y → C (x * y)) :

C z

def free_comm_ring.lift {α : Type u} {R : Type v} [comm_ring R] (f : α → R) :

free_comm_ring α →+* R

Lift a map α → R to a additive group homomorphism free_comm_ring α → R. For a version producing a bundled homomorphism, see lift_hom.

Equations

free_comm_ring.lift f = {to_fun := (free_abelian_group.lift (λ (s : multiplicative (multiset α)), (multiset.map f s.to_add).prod)).to_fun, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

@[simp]

theorem free_comm_ring.lift_of {α : Type u} {R : Type v} [comm_ring R] (f : α → R) (x : α) :

⇑(free_comm_ring.lift f) (free_comm_ring.of x) = f x

@[simp]

theorem free_comm_ring.lift_comp_of {α : Type u} {R : Type v} [comm_ring R] (f : free_comm_ring α →+* R) :

free_comm_ring.lift (⇑f ∘ free_comm_ring.of) = f

def free_comm_ring.map {α : Type u} {β : Type v} (f : α → β) :

free_comm_ring α →+* free_comm_ring β

A map f : α → β produces a ring homomorphism free_comm_ring α →+* free_comm_ring β.

Equations

free_comm_ring.map f = free_comm_ring.lift (free_comm_ring.of ∘ f)

@[simp]

theorem free_comm_ring.map_of {α : Type u} {β : Type v} (f : α → β) (x : α) :

⇑(free_comm_ring.map f) (free_comm_ring.of x) = free_comm_ring.of (f x)

def free_comm_ring.is_supported {α : Type u} (x : free_comm_ring α) (s : set α) :

Prop

is_supported x s means that all monomials showing up in x have variables in s.

Equations

x.is_supported s = (x ∈ ring.closure (free_comm_ring.of '' s))

theorem free_comm_ring.is_supported_upwards {α : Type u} {x : free_comm_ring α} {s t : set α} (hs : x.is_supported s) (hst : s ⊆ t) :

x.is_supported t

theorem free_comm_ring.is_supported_add {α : Type u} {x y : free_comm_ring α} {s : set α} (hxs : x.is_supported s) (hys : y.is_supported s) :

(x + y).is_supported s

theorem free_comm_ring.is_supported_neg {α : Type u} {x : free_comm_ring α} {s : set α} (hxs : x.is_supported s) :

(-x).is_supported s

theorem free_comm_ring.is_supported_sub {α : Type u} {x y : free_comm_ring α} {s : set α} (hxs : x.is_supported s) (hys : y.is_supported s) :

(x - y).is_supported s

theorem free_comm_ring.is_supported_mul {α : Type u} {x y : free_comm_ring α} {s : set α} (hxs : x.is_supported s) (hys : y.is_supported s) :

(x * y).is_supported s

theorem free_comm_ring.is_supported_zero {α : Type u} {s : set α} :

0.is_supported s

theorem free_comm_ring.is_supported_one {α : Type u} {s : set α} :

1.is_supported s

theorem free_comm_ring.is_supported_int {α : Type u} {i : ℤ} {s : set α} :

↑i.is_supported s

def free_comm_ring.restriction {α : Type u} (s : set α) [decidable_pred s] :

free_comm_ring α →+* free_comm_ring ↥s

The restriction map from free_comm_ring α to free_comm_ring s where s : set α, defined by sending all variables not in s to zero.

Equations

free_comm_ring.restriction s = free_comm_ring.lift (λ (p : α), dite (p ∈ s) (λ (H : p ∈ s), free_comm_ring.of ⟨p, H⟩) (λ (H : p ∉ s), 0))

@[simp]

theorem free_comm_ring.restriction_of {α : Type u} (s : set α) [decidable_pred s] (p : α) :

⇑(free_comm_ring.restriction s) (free_comm_ring.of p) = dite (p ∈ s) (λ (H : p ∈ s), free_comm_ring.of ⟨p, H⟩) (λ (H : p ∉ s), 0)

theorem free_comm_ring.is_supported_of {α : Type u} {p : α} {s : set α} :

(free_comm_ring.of p).is_supported s ↔ p ∈ s

theorem free_comm_ring.map_subtype_val_restriction {α : Type u} {x : free_comm_ring α} (s : set α) [decidable_pred s] (hxs : x.is_supported s) :

⇑(free_comm_ring.map subtype.val) (⇑(free_comm_ring.restriction s) x) = x

theorem free_comm_ring.exists_finite_support {α : Type u} (x : free_comm_ring α) :

∃ (s : set α), s.finite ∧ x.is_supported s

theorem free_comm_ring.exists_finset_support {α : Type u} (x : free_comm_ring α) :

∃ (s : finset α), x.is_supported ↑s

def free_ring.to_free_comm_ring {α : Type u_1} :

free_ring α →+* free_comm_ring α

The canonical ring homomorphism from the free ring generated by α to the free commutative ring generated by α.

Equations

free_ring.to_free_comm_ring = free_ring.lift free_comm_ring.of

@[instance]

def free_ring.has_coe (α : Type u) :

has_coe (free_ring α) (free_comm_ring α)

Equations

free_ring.has_coe α = {coe := ⇑free_ring.to_free_comm_ring}

@[instance]

def free_ring.coe.is_ring_hom (α : Type u) :

is_ring_hom coe

@[simp]

theorem free_ring.coe_zero (α : Type u) :

↑0 = 0

@[simp]

theorem free_ring.coe_one (α : Type u) :

↑1 = 1

@[simp]

theorem free_ring.coe_of {α : Type u} (a : α) :

↑(free_ring.of a) = free_comm_ring.of a

@[simp]

theorem free_ring.coe_neg {α : Type u} (x : free_ring α) :

@[simp]

theorem free_ring.coe_add {α : Type u} (x y : free_ring α) :

↑(x + y) = ↑x + ↑y

@[simp]

theorem free_ring.coe_sub {α : Type u} (x y : free_ring α) :

↑(x - y) = ↑x - ↑y

@[simp]

theorem free_ring.coe_mul {α : Type u} (x y : free_ring α) :

↑x * y = (↑x) * ↑y

theorem free_ring.coe_surjective (α : Type u) :

function.surjective coe

theorem free_ring.coe_eq (α : Type u) :

coe = functor.map (λ (l : list α), ↑l)

def free_ring.of' {R : Type u_1} {S : Type u_2} [ring R] [ring S] (e : R ≃ S) [is_ring_hom ⇑e] :

Interpret an equivalence f : R ≃ S as a ring equivalence R ≃+* S.

Equations

free_ring.of' e = {to_fun := e.to_fun, inv_fun := e.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

def free_ring.subsingleton_equiv_free_comm_ring (α : Type u) [subsingleton α] :

free_ring α ≃+* free_comm_ring α

If α has size at most 1 then the natural map from the free ring on α to the free commutative ring on α is an isomorphism of rings.

Equations

free_ring.subsingleton_equiv_free_comm_ring α = free_ring.of' (functor.map_equiv free_abelian_group (multiset.subsingleton_equiv α))

@[instance]

def free_ring.comm_ring (α : Type u) [subsingleton α] :

comm_ring (free_ring α)

Equations

free_ring.comm_ring α = {add := ring.add (free_ring.ring α), add_assoc := _, zero := ring.zero (free_ring.ring α), zero_add := _, add_zero := _, neg := ring.neg (free_ring.ring α), add_left_neg := _, add_comm := _, mul := ring.mul (free_ring.ring α), mul_assoc := _, one := ring.one (free_ring.ring α), one_mul := _, mul_one := _, left_distrib := _, right_distrib := _, mul_comm := _}

def free_comm_ring_equiv_mv_polynomial_int (α : Type u) :

free_comm_ring α ≃+* mv_polynomial α ℤ

The free commutative ring on α is isomorphic to the polynomial ring over ℤ with variables in α

Equations

free_comm_ring_equiv_mv_polynomial_int α = {to_fun := ⇑(free_comm_ring.lift (λ (a : α), mv_polynomial.X a)), inv_fun := mv_polynomial.eval₂ (int.cast_ring_hom (free_comm_ring α)) free_comm_ring.of, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

def free_comm_ring_pempty_equiv_int :

free_comm_ring pempty ≃+* ℤ

The free commutative ring on the empty type is isomorphic to ℤ.

Equations

free_comm_ring_pempty_equiv_int = (free_comm_ring_equiv_mv_polynomial_int pempty).trans (mv_polynomial.pempty_ring_equiv ℤ)

def free_comm_ring_punit_equiv_polynomial_int :

free_comm_ring punit ≃+* polynomial ℤ

The free commutative ring on a type with one term is isomorphic to ℤ[X].

Equations

free_comm_ring_punit_equiv_polynomial_int = (free_comm_ring_equiv_mv_polynomial_int punit).trans (mv_polynomial.punit_ring_equiv ℤ)

def free_ring_pempty_equiv_int :

free_ring pempty ≃+* ℤ

The free ring on the empty type is isomorphic to ℤ.

Equations

free_ring_pempty_equiv_int = (free_ring.subsingleton_equiv_free_comm_ring pempty).trans free_comm_ring_pempty_equiv_int

def free_ring_punit_equiv_polynomial_int :

free_ring punit ≃+* polynomial ℤ

The free ring on a type with one term is isomorphic to ℤ[X].

Equations

free_ring_punit_equiv_polynomial_int = (free_ring.subsingleton_equiv_free_comm_ring punit).trans free_comm_ring_punit_equiv_polynomial_int