Theory RingHom

(*  Title:      HOL/Algebra/RingHom.thy
    Author:     Stephan Hohe, TU Muenchen
*)

theory RingHom
imports Ideal
begin

section Homomorphisms of Non-Commutative Rings

text Lifting existing lemmas in a ring_hom_ring› locale
locale ring_hom_ring = R?: ring R + S?: ring S
    for R (structure) and S (structure) +
  fixes h
  assumes homh: "h  ring_hom R S"
  notes hom_mult [simp] = ring_hom_mult [OF homh]
    and hom_one [simp] = ring_hom_one [OF homh]

sublocale ring_hom_cring  ring: ring_hom_ring
  by standard (rule homh)

sublocale ring_hom_ring  abelian_group?: abelian_group_hom R S
proof 
  show "h  hom (add_monoid R) (add_monoid S)"
    using homh by (simp add: hom_def ring_hom_def)
qed

lemma (in ring_hom_ring) is_ring_hom_ring:
  "ring_hom_ring R S h"
  by (rule ring_hom_ring_axioms)

lemma ring_hom_ringI:
  fixes R (structure) and S (structure)
  assumes "ring R" "ring S"
  assumes hom_closed: "!!x. x  carrier R ==> h x  carrier S"
      and compatible_mult: "x y. [| x  carrier R; y  carrier R |] ==> h (x  y) = h x Sh y"
      and compatible_add: "x y. [| x  carrier R; y  carrier R |] ==> h (x  y) = h x Sh y"
      and compatible_one: "h 𝟭 = 𝟭S"
  shows "ring_hom_ring R S h"
proof -
  interpret ring R by fact
  interpret ring S by fact
  show ?thesis
  proof
    show "h  ring_hom R S"
      unfolding ring_hom_def
      by (auto simp: compatible_mult compatible_add compatible_one hom_closed)
  qed
qed

lemma ring_hom_ringI2:
  assumes "ring R" "ring S"
  assumes h: "h  ring_hom R S"
  shows "ring_hom_ring R S h"
proof -
  interpret R: ring R by fact
  interpret S: ring S by fact
  show ?thesis 
  proof
    show "h  ring_hom R S"
      using h .
  qed
qed

lemma ring_hom_ringI3:
  fixes R (structure) and S (structure)
  assumes "abelian_group_hom R S h" "ring R" "ring S" 
  assumes compatible_mult: "x y. [| x  carrier R; y  carrier R |] ==> h (x  y) = h x Sh y"
      and compatible_one: "h 𝟭 = 𝟭S"
  shows "ring_hom_ring R S h"
proof -
  interpret abelian_group_hom R S h by fact
  interpret R: ring R by fact
  interpret S: ring S by fact
  show ?thesis
  proof
    show "h  ring_hom R S"
      unfolding ring_hom_def by (auto simp: compatible_one compatible_mult)
  qed
qed

lemma ring_hom_cringI:
  assumes "ring_hom_ring R S h" "cring R" "cring S"
  shows "ring_hom_cring R S h"
proof -
  interpret ring_hom_ring R S h by fact
  interpret R: cring R by fact
  interpret S: cring S by fact
  show ?thesis 
  proof
    show "h  ring_hom R S"
      by (simp add: homh)
  qed
qed


subsection The Kernel of a Ring Homomorphism

― ‹the kernel of a ring homomorphism is an ideal›
lemma (in ring_hom_ring) kernel_is_ideal: "ideal (a_kernel R S h) R"
  apply (rule idealI [OF R.is_ring])
    apply (rule additive_subgroup.a_subgroup[OF additive_subgroup_a_kernel])
   apply (auto simp: a_kernel_def')
  done

text Elements of the kernel are mapped to zero
lemma (in abelian_group_hom) kernel_zero [simp]:
  "i  a_kernel R S h  h i = 𝟬S"
by (simp add: a_kernel_defs)


subsection Cosets

text Cosets of the kernel correspond to the elements of the image of the homomorphism
lemma (in ring_hom_ring) rcos_imp_homeq:
  assumes acarr: "a  carrier R"
      and xrcos: "x  a_kernel R S h +> a"
  shows "h x = h a"
proof -
  interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)

  from xrcos
      have "i  a_kernel R S h. x = i  a" by (simp add: a_r_coset_defs)
  from this obtain i
      where iker: "i  a_kernel R S h"
        and x: "x = i  a"
      by fast+
  note carr = acarr iker[THEN a_Hcarr]

  from x
      have "h x = h (i  a)" by simp
  also from carr
      have " = h i Sh a" by simp
  also from iker
      have " = 𝟬SSh a" by simp
  also from carr
      have " = h a" by simp
  finally
      show "h x = h a" .
qed

lemma (in ring_hom_ring) homeq_imp_rcos:
  assumes acarr: "a  carrier R"
      and xcarr: "x  carrier R"
      and hx: "h x = h a"
  shows "x  a_kernel R S h +> a"
proof -
  interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)
 
  note carr = acarr xcarr
  note hcarr = acarr[THEN hom_closed] xcarr[THEN hom_closed]

  from hx and hcarr
      have a: "h x SSh a = 𝟬S" by algebra
  from carr
      have "h x SSh a = h (x  a)" by simp
  from a and this
      have b: "h (x  a) = 𝟬S" by simp

  from carr have "x  a  carrier R" by simp
  from this and b
      have "x  a  a_kernel R S h" 
      unfolding a_kernel_def'
      by fast

  from this and carr
      show "x  a_kernel R S h +> a" by (simp add: a_rcos_module_rev)
qed

corollary (in ring_hom_ring) rcos_eq_homeq:
  assumes acarr: "a  carrier R"
  shows "(a_kernel R S h) +> a = {x  carrier R. h x = h a}"
proof -
  interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)
  show ?thesis
    using assms by (auto simp: intro: homeq_imp_rcos rcos_imp_homeq a_elemrcos_carrier)
qed

lemma (in ring_hom_ring) hom_nat_pow:
  "x  carrier R  h (x [^] (n :: nat)) = (h x) [^]Sn"
  by (induct n) (auto)


lemma (in ring_hom_ring) inj_on_domain: contributor Paulo Emílio de Vilhena
  assumes "inj_on h (carrier R)"
  shows "domain S  domain R"
proof -
  assume A: "domain S" show "domain R"
  proof
    have "h 𝟭 = 𝟭S h 𝟬 = 𝟬S" by simp
    hence "h 𝟭  h 𝟬"
      using domain.one_not_zero[OF A] by simp
    thus "𝟭  𝟬"
      using assms unfolding inj_on_def by fastforce 
  next
    fix a b
    assume a: "a  carrier R"
       and b: "b  carrier R"
    have "h (a  b) = (h a) S(h b)" by (simp add: a b)
    also have " ... = (h b) S(h a)" using a b A cringE(1)[of S]
      by (simp add: cring.cring_simprules(14) domain_def)
    also have " ... = h (b  a)" by (simp add: a b)
    finally have "h (a  b) = h (b  a)" .
    thus "a  b = b  a"
      using assms a b unfolding inj_on_def by simp 
    
    assume  ab: "a  b = 𝟬"
    hence "h (a  b) = 𝟬S" by simp
    hence "(h a) S(h b) = 𝟬S" using a b by simp
    hence "h a =  𝟬S h b =  𝟬S" using a b domain.integral[OF A] by simp
    thus "a = 𝟬  b = 𝟬"
      using a b assms unfolding inj_on_def by force
  qed
qed

end