Documentation

Mathlib.Algebra.Ring.Prod

Semiring, ring etc structures on R × S #

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 RingHoms and similarly for NonUnitalRingHoms:

instance Prod.instDistrib {R : Type u_1} {S : Type u_3} [Distrib R] [Distrib S] :
Distrib (R × S)

Product of two distributive types is distributive.

Equations
instance Prod.instSemiring {R : Type u_1} {S : Type u_3} [Semiring R] [Semiring S] :
Semiring (R × S)

Product of two semirings is a semiring.

Equations
instance Prod.instCommSemiring {R : Type u_1} {S : Type u_3} [CommSemiring R] [CommSemiring S] :

Product of two commutative semirings is a commutative semiring.

Equations
instance Prod.instNonAssocRing {R : Type u_1} {S : Type u_3} [NonAssocRing R] [NonAssocRing S] :
Equations
instance Prod.instRing {R : Type u_1} {S : Type u_3} [Ring R] [Ring S] :
Ring (R × S)

Product of two rings is a ring.

Equations
instance Prod.instCommRing {R : Type u_1} {S : Type u_3} [CommRing R] [CommRing S] :
CommRing (R × S)

Product of two commutative rings is a commutative ring.

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

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
  • f.prod g = { toFun := fun (x : R) => (f x, g x), map_mul' := , map_zero' := , map_add' := }
@[simp]
theorem NonUnitalRingHom.prod_apply {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) (x : R) :
(f.prod g) x = (f x, g x)
@[simp]
theorem NonUnitalRingHom.fst_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) :
(fst S T).comp (f.prod g) = f
@[simp]
theorem NonUnitalRingHom.snd_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) :
(snd S T).comp (f.prod g) = g
theorem NonUnitalRingHom.prod_unique {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S × T) :
((fst S T).comp f).prod ((snd S T).comp f) = f

Prod.map as a NonUnitalRingHom.

Equations
theorem NonUnitalRingHom.prodMap_def {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') :
f.prodMap g = (f.comp (fst R S)).prod (g.comp (snd R S))
@[simp]
theorem NonUnitalRingHom.coe_prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') :
(f.prodMap g) = Prod.map f g
theorem NonUnitalRingHom.prod_comp_prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} {T : Type u_5} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S'] [NonUnitalNonAssocSemiring T] (f : T →ₙ+* R) (g : T →ₙ+* S) (f' : R →ₙ+* R') (g' : S →ₙ+* S') :
(f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
def RingHom.fst (R : Type u_1) (S : Type u_3) [NonAssocSemiring R] [NonAssocSemiring S] :
R × S →+* R

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

Equations
  • RingHom.fst R S = { toFun := Prod.fst, map_one' := , map_mul' := , map_zero' := , map_add' := }
def RingHom.snd (R : Type u_1) (S : Type u_3) [NonAssocSemiring R] [NonAssocSemiring S] :
R × S →+* S

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

Equations
  • RingHom.snd R S = { toFun := Prod.snd, map_one' := , map_mul' := , map_zero' := , map_add' := }
@[simp]
theorem RingHom.coe_fst {R : Type u_1} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] :
(fst R S) = Prod.fst
@[simp]
theorem RingHom.coe_snd {R : Type u_1} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] :
(snd R S) = Prod.snd
def RingHom.prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring 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
  • f.prod g = { toFun := fun (x : R) => (f x, g x), map_one' := , map_mul' := , map_zero' := , map_add' := }
@[simp]
theorem RingHom.prod_apply {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) (x : R) :
(f.prod g) x = (f x, g x)
@[simp]
theorem RingHom.fst_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) :
(fst S T).comp (f.prod g) = f
@[simp]
theorem RingHom.snd_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) :
(snd S T).comp (f.prod g) = g
theorem RingHom.prod_unique {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S × T) :
((fst S T).comp f).prod ((snd S T).comp f) = f
def RingHom.prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') :
R × S →+* R' × S'

Prod.map as a RingHom.

Equations
theorem RingHom.prodMap_def {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') :
f.prodMap g = (f.comp (fst R S)).prod (g.comp (snd R S))
@[simp]
theorem RingHom.coe_prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') :
(f.prodMap g) = Prod.map f g
theorem RingHom.prod_comp_prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] [NonAssocSemiring T] (f : T →+* R) (g : T →+* S) (f' : R →+* R') (g' : S →+* S') :
(f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
def RingEquiv.prodComm {R : Type u_1} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] :
R × S ≃+* S × R

Swapping components as an equivalence of (semi)rings.

Equations
def RingEquiv.prodProdProdComm (R : Type u_1) (R' : Type u_2) (S : Type u_3) (S' : Type u_4) [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] :
(R × R') × S × S' ≃+* (R × S) × R' × S'

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

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem RingEquiv.prodProdProdComm_apply (R : Type u_1) (R' : Type u_2) (S : Type u_3) (S' : Type u_4) [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (rrss : (R × R') × S × S') :
(prodProdProdComm R R' S S') rrss = ((rrss.1.1, rrss.2.1), rrss.1.2, rrss.2.2)
@[simp]
theorem RingEquiv.prodProdProdComm_symm (R : Type u_1) (R' : Type u_2) (S : Type u_3) (S' : Type u_4) [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] :
(prodProdProdComm R R' S S').symm = prodProdProdComm R S R' S'
@[simp]
@[simp]
@[simp]
theorem RingEquiv.prodProdProdComm_toEquiv (R : Type u_1) (R' : Type u_2) (S : Type u_3) (S' : Type u_4) [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] :
(prodProdProdComm R R' S S') = Equiv.prodProdProdComm R R' S S'

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

Equations
@[simp]
theorem RingEquiv.prodZeroRing_apply (R : Type u_1) (S : Type u_3) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (x : R) :
(prodZeroRing R S) x = (x, 0)
@[simp]
theorem RingEquiv.prodZeroRing_symm_apply (R : Type u_1) (S : Type u_3) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (self : R × S) :
(prodZeroRing R S).symm self = self.1

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

Equations
@[simp]
theorem RingEquiv.zeroRingProd_symm_apply (R : Type u_1) (S : Type u_3) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (self : S × R) :
(zeroRingProd R S).symm self = self.2
@[simp]
theorem RingEquiv.zeroRingProd_apply (R : Type u_1) (S : Type u_3) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (x : R) :
(zeroRingProd R S) x = (0, x)
theorem false_of_nontrivial_of_product_domain (R : Type u_6) (S : Type u_7) [Ring R] [Ring S] [IsDomain (R × S)] [Nontrivial R] [Nontrivial S] :

The product of two nontrivial rings is not a domain