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 @{text 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 default (rule homh)

sublocale ring_hom_ring ⊆ abelian_group: abelian_group_hom R S
apply (rule abelian_group_homI)
  apply (rule R.is_abelian_group)
 apply (rule S.is_abelian_group)
apply (intro group_hom.intro group_hom_axioms.intro)
  apply (rule R.a_group)
 apply (rule S.a_group)
apply (insert homh, unfold hom_def ring_hom_def)
apply simp
done

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 (* morphism: "h ∈ carrier R -> carrier S" *)
          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 ⊗⇘S⇙ h y"
      and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x ⊕ y) = h x ⊕⇘S⇙ h y"
      and compatible_one: "h \<one> = \<one>⇘S⇙"
  shows "ring_hom_ring R S h"
proof -
  interpret ring R by fact
  interpret ring S by fact
  show ?thesis apply unfold_locales
apply (unfold ring_hom_def, safe)
   apply (simp add: hom_closed Pi_def)
  apply (erule (1) compatible_mult)
 apply (erule (1) compatible_add)
apply (rule compatible_one)
done
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 apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro)
    apply (rule R.is_ring)
    apply (rule S.is_ring)
    apply (rule h)
    done
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 ⊗⇘S⇙ h y"
      and compatible_one: "h \<one> = \<one>⇘S⇙"
  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 apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro, rule R.is_ring, rule S.is_ring)
    apply (insert group_hom.homh[OF a_group_hom])
    apply (unfold hom_def ring_hom_def, simp)
    apply safe
    apply (erule (1) compatible_mult)
    apply (rule compatible_one)
    done
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 by (intro ring_hom_cring.intro ring_hom_cring_axioms.intro)
    (rule R.is_cring, rule S.is_cring, rule homh)
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:
  shows "ideal (a_kernel R S h) R"
apply (rule idealI)
   apply (rule R.is_ring)
  apply (rule additive_subgroup.a_subgroup[OF additive_subgroup_a_kernel])
 apply (unfold a_kernel_def', simp+)
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 = \<zero>⇘S⇙"
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 ⊕⇘S⇙ h a" by simp
  also from iker
      have "… = \<zero>⇘S⇙ ⊕⇘S⇙ h 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 ⊕⇘S⇙ \<ominus>⇘S⇙h a = \<zero>⇘S⇙" by algebra
  from carr
      have "h x ⊕⇘S⇙ \<ominus>⇘S⇙h a = h (x ⊕ \<ominus>a)" by simp
  from a and this
      have b: "h (x ⊕ \<ominus>a) = \<zero>⇘S⇙" by simp

  from carr have "x ⊕ \<ominus>a ∈ carrier R" by simp
  from this and b
      have "x ⊕ \<ominus>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}"
apply rule defer 1
apply clarsimp defer 1
proof
  interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)

  fix x
  assume xrcos: "x ∈ a_kernel R S h +> a"
  from acarr and this
      have xcarr: "x ∈ carrier R"
      by (rule a_elemrcos_carrier)

  from xrcos
      have "h x = h a" by (rule rcos_imp_homeq[OF acarr])
  from xcarr and this
      show "x ∈ {x ∈ carrier R. h x = h a}" by fast
next
  interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)

  fix x
  assume xcarr: "x ∈ carrier R"
     and hx: "h x = h a"
  from acarr xcarr hx
      show "x ∈ a_kernel R S h +> a" by (rule homeq_imp_rcos)
qed

end