mathlib documentation

ring_theory.free_ring

Free rings #

The theory of the free ring over a type.

Main definitions #

Implementation details #

free_ring α is implemented as the free abelian group over the free monoid on α.

Tags #

free ring

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def free_ring (α : Type u) :
Type u

The free ring over a type α.

Equations
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@[protected, instance]
def free_ring.inhabited (α : Type u) :
inhabited (free_ring α)
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@[protected, instance]
def free_ring.ring (α : Type u) :
ring (free_ring α)
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def free_ring.of {α : Type u} (x : α) :
free_ring α

The canonical map from α to free_ring α.

Equations
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theorem free_ring.of_injective {α : Type u} :
function.injective free_ring.of
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@[protected]
theorem free_ring.induction_on {α : Type u} {C : free_ring α → Prop} (z : free_ring α) (hn1 : C (-1)) (hb : ∀ (b : α), C (free_ring.of b)) (ha : ∀ (x y : free_ring α), C x → C y → C (x + y)) (hm : ∀ (x y : free_ring α), C x → C y → C (x * y)) :
C z
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def free_ring.lift {α : Type u} {R : Type v} [ring R] :
(α → R) ≃ (free_ring α →+* R)

The ring homomorphism free_ring α →+* R induced from a map α → R.

Equations
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@[simp]
theorem free_ring.lift_of {α : Type u} {R : Type v} [ring R] (f : α → R) (x : α) :
⇑(⇑free_ring.lift f) (free_ring.of x) = f x
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@[simp]
theorem free_ring.lift_comp_of {α : Type u} {R : Type v} [ring R] (f : free_ring α →+* R) :
⇑free_ring.lift (⇑f ∘ free_ring.of) = f
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@[ext]
theorem free_ring.hom_ext {α : Type u} {R : Type v} [ring R] ⦃f g : free_ring α →+* R⦄ (h : ∀ (x : α), ⇑f (free_ring.of x) = ⇑g (free_ring.of x)) :
f = g
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def free_ring.map {α : Type u} {β : Type v} (f : α → β) :
free_ring α →+* free_ring β

The canonical ring homomorphism free_ring α →+* free_ring β generated by a map α → β.

Equations
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@[simp]
theorem free_ring.map_of {α : Type u} {β : Type v} (f : α → β) (x : α) :
⇑(free_ring.map f) (free_ring.of x) = free_ring.of (f x)