mathlib3 documentation

algebra.ring.prod

Semiring, ring etc structures on R × S #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file we define two-binop (semiring, ring etc) structures on R × S. We also prove trivial simp lemmas, and define the following operations on ring_homs and similarly for non_unital_ring_homs:

@[protected, instance]
def prod.distrib {R : Type u_3} {S : Type u_5} [distrib R] [distrib S] :
distrib (R × S)

Product of two distributive types is distributive.

Equations
@[protected, instance]
def prod.comm_semiring {R : Type u_3} {S : Type u_5} [comm_semiring R] [comm_semiring S] :

Product of two commutative semirings is a commutative semiring.

Equations

Given non-unital semirings R, S, the natural projection homomorphism from R × S to R.

Equations

Given non-unital semirings R, S, the natural projection homomorphism from R × S to S.

Equations
@[protected]

Combine two non-unital ring homomorphisms f : R →ₙ+* S, g : R →ₙ+* T into f.prod g : R →ₙ+* S × T given by (f.prod g) x = (f x, g x)

Equations
@[simp]
theorem non_unital_ring_hom.prod_apply {R : Type u_3} {S : Type u_5} {T : Type u_7} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_non_assoc_semiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) (x : R) :
(f.prod g) x = (f x, g x)
theorem non_unital_ring_hom.prod_comp_prod_map {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} {T : Type u_7} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [non_unital_non_assoc_semiring R'] [non_unital_non_assoc_semiring S'] [non_unital_non_assoc_semiring T] (f : T →ₙ+* R) (g : T →ₙ+* S) (f' : R →ₙ+* R') (g' : S →ₙ+* S') :
(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
def ring_hom.fst (R : Type u_3) (S : Type u_5) [non_assoc_semiring R] [non_assoc_semiring S] :
R × S →+* R

Given semirings R, S, the natural projection homomorphism from R × S to R.

Equations
def ring_hom.snd (R : Type u_3) (S : Type u_5) [non_assoc_semiring R] [non_assoc_semiring S] :
R × S →+* S

Given semirings R, S, the natural projection homomorphism from R × S to S.

Equations
@[simp]
@[simp]
@[protected]
def ring_hom.prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (f : R →+* S) (g : R →+* T) :
R →+* S × T

Combine two ring homomorphisms f : R →+* S, g : R →+* T into f.prod g : R →+* S × T given by (f.prod g) x = (f x, g x)

Equations
@[simp]
theorem ring_hom.prod_apply {R : Type u_3} {S : Type u_5} {T : Type u_7} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (f : R →+* S) (g : R →+* T) (x : R) :
(f.prod g) x = (f x, g x)
@[simp]
theorem ring_hom.fst_comp_prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (f : R →+* S) (g : R →+* T) :
(ring_hom.fst S T).comp (f.prod g) = f
@[simp]
theorem ring_hom.snd_comp_prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (f : R →+* S) (g : R →+* T) :
(ring_hom.snd S T).comp (f.prod g) = g
theorem ring_hom.prod_unique {R : Type u_3} {S : Type u_5} {T : Type u_7} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (f : R →+* S × T) :
((ring_hom.fst S T).comp f).prod ((ring_hom.snd S T).comp f) = f
def ring_hom.prod_map {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring R'] [non_assoc_semiring S'] (f : R →+* R') (g : S →+* S') :
R × S →+* R' × S'

prod.map as a ring_hom.

Equations
theorem ring_hom.prod_map_def {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring R'] [non_assoc_semiring S'] (f : R →+* R') (g : S →+* S') :
f.prod_map g = (f.comp (ring_hom.fst R S)).prod (g.comp (ring_hom.snd R S))
@[simp]
theorem ring_hom.coe_prod_map {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring R'] [non_assoc_semiring S'] (f : R →+* R') (g : S →+* S') :
theorem ring_hom.prod_comp_prod_map {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} {T : Type u_7} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring R'] [non_assoc_semiring S'] [non_assoc_semiring T] (f : T →+* R) (g : T →+* S) (f' : R →+* R') (g' : S →+* S') :
(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
def ring_equiv.prod_comm {R : Type u_3} {S : Type u_5} [non_assoc_semiring R] [non_assoc_semiring S] :
R × S ≃+* S × R

Swapping components as an equivalence of (semi)rings.

Equations
@[simp]
theorem ring_equiv.prod_prod_prod_comm_apply (R : Type u_3) (R' : Type u_4) (S : Type u_5) (S' : Type u_6) [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring R'] [non_assoc_semiring S'] (rrss : (R × R') × S × S') :
(ring_equiv.prod_prod_prod_comm R R' S S') rrss = ((rrss.fst.fst, rrss.snd.fst), rrss.fst.snd, rrss.snd.snd)
def ring_equiv.prod_prod_prod_comm (R : Type u_3) (R' : Type u_4) (S : Type u_5) (S' : Type u_6) [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring R'] [non_assoc_semiring S'] :
(R × R') × S × S' ≃+* (R × S) × R' × S'

Four-way commutativity of prod. The name matches mul_mul_mul_comm.

Equations

A ring R is isomorphic to R × S when S is the zero ring

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@[simp]
@[simp]

A ring R is isomorphic to S × R when S is the zero ring

Equations
@[simp]
@[simp]
theorem false_of_nontrivial_of_product_domain (R : Type u_1) (S : Type u_2) [ring R] [ring S] [is_domain (R × S)] [nontrivial R] [nontrivial S] :

The product of two nontrivial rings is not a domain

Order #