Theory RingHom

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theory RingHom
imports Ideal
begin

(*
  Title:     HOL/Algebra/RingHom.thy
  Id:        $Id: RingHom.thy,v 1.5 2007/06/21 18:07:28 wenzelm Exp $
  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 = ring R + ring S + var 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]

interpretation ring_hom_cring ⊆ ring_hom_ring
  by (unfold_locales, rule homh)

interpretation ring_hom_ring ⊆ 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:
  includes struct R + struct S
  shows "ring_hom_ring R S h"
by fact

lemma ring_hom_ringI:
  includes 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"
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

lemma ring_hom_ringI2:
  includes ring R + ring S
  assumes h: "h ∈ ring_hom R S"
  shows "ring_hom_ring R S h"
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

lemma ring_hom_ringI3:
  includes abelian_group_hom R S + 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"
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

lemma ring_hom_cringI:
  includes ring_hom_ring R S h + cring R + cring S
  shows "ring_hom_cring R S h"
  by (intro ring_hom_cring.intro ring_hom_cring_axioms.intro)
    (rule R.is_cring, rule S.is_cring, rule homh)


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>SS 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>Sh a = \<zero>S" by algebra
  from carr
      have "h x ⊕S \<ominus>Sh 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

Homomorphisms of Non-Commutative Rings

lemma is_ring_hom_ring:

  ring_hom_ring R S h

lemma ring_hom_ringI:

  [| Ring.ring R; Ring.ring S; !!x. x ∈ carrier R ==> h x ∈ carrier S;
     !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> h (xR y) = h xS h y;
     !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> h (xR y) = h xS h y;
     h \<one>R = \<one>S |]
  ==> ring_hom_ring R S h

lemma ring_hom_ringI2:

  [| Ring.ring R; Ring.ring S; h ∈ ring_hom R S |] ==> ring_hom_ring R S h

lemma ring_hom_ringI3:

  [| abelian_group_hom R S h; Ring.ring R; Ring.ring S;
     !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> h (xR y) = h xS h y;
     h \<one>R = \<one>S |]
  ==> ring_hom_ring R S h

lemma ring_hom_cringI:

  [| ring_hom_ring R S h; cring R; cring S |] ==> ring_hom_cring R S h

The kernel of a ring homomorphism

lemma kernel_is_ideal:

  ideal (a_kernel R S h) R

lemma kernel_zero:

  i ∈ a_kernel R S h ==> h i = \<zero>S

Cosets

lemma rcos_imp_homeq:

  [| a ∈ carrier R; x ∈ a_kernel R S h +> a |] ==> h x = h a

lemma homeq_imp_rcos:

  [| a ∈ carrier R; x ∈ carrier R; h x = h a |] ==> x ∈ a_kernel R S h +> a

corollary rcos_eq_homeq:

  a ∈ carrier R ==> a_kernel R S h +> a = {x : carrier R. h x = h a}