src/HOL/Algebra/RingHom.thy
changeset 20318 0e0ea63fe768
child 21502 7f3ea2b3bab6
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Algebra/RingHom.thy	Thu Aug 03 14:57:26 2006 +0200
@@ -0,0 +1,178 @@
+(*
+  Title:     HOL/Algebra/RingHom.thy
+  Id:        $Id$
+  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 = ring R + ring S + var h +
+  assumes homh: "h \<in> 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 \<subseteq> ring_hom_ring
+  by (unfold_locales, rule homh)
+
+interpretation ring_hom_ring \<subseteq> 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 -
+
+lemma ring_hom_ringI:
+  includes ring R + ring S
+  assumes (* morphism: "h \<in> carrier R \<rightarrow> carrier S" *)
+          hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
+      and compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
+      and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
+      and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
+  shows "ring_hom_ring R S h"
+apply unfold_locales
+apply (unfold ring_hom_def, safe)
+   apply (simp add: hom_closed Pi_def, assumption+)
+done
+
+lemma ring_hom_ringI2:
+  includes ring R + ring S
+  assumes "h \<in> ring_hom R S"
+  shows "ring_hom_ring R S h"
+by (intro ring_hom_ring.intro ring_hom_ring_axioms.intro)
+
+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 \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
+      and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
+  shows "ring_hom_ring R S h"
+apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro, assumption+)
+apply (insert group_hom.homh[OF a_group_hom])
+apply (unfold hom_def ring_hom_def, simp)
+apply (safe, assumption+)
+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, assumption+, 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 \<in> a_kernel R S h \<Longrightarrow> h i = \<zero>\<^bsub>S\<^esub>"
+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 \<in> carrier R"
+      and xrcos: "x \<in> 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 "\<exists>i \<in> a_kernel R S h. x = i \<oplus> a" by (simp add: a_r_coset_defs)
+  from this obtain i
+      where iker: "i \<in> a_kernel R S h"
+        and x: "x = i \<oplus> a"
+      by fast+
+  note carr = acarr iker[THEN a_Hcarr]
+
+  from x
+      have "h x = h (i \<oplus> a)" by simp
+  also from carr
+      have "\<dots> = h i \<oplus>\<^bsub>S\<^esub> h a" by simp
+  also from iker
+      have "\<dots> = \<zero>\<^bsub>S\<^esub> \<oplus>\<^bsub>S\<^esub> h a" by simp
+  also from carr
+      have "\<dots> = h a" by simp
+  finally
+      show "h x = h a" .
+qed
+
+lemma (in ring_hom_ring) homeq_imp_rcos:
+  assumes acarr: "a \<in> carrier R"
+      and xcarr: "x \<in> carrier R"
+      and hx: "h x = h a"
+  shows "x \<in> 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 \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = \<zero>\<^bsub>S\<^esub>" by algebra
+  from carr
+      have "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = h (x \<oplus> \<ominus>a)" by simp
+  from a and this
+      have b: "h (x \<oplus> \<ominus>a) = \<zero>\<^bsub>S\<^esub>" by simp
+
+  from carr have "x \<oplus> \<ominus>a \<in> carrier R" by simp
+  from this and b
+      have "x \<oplus> \<ominus>a \<in> a_kernel R S h" 
+      unfolding a_kernel_def'
+      by fast
+
+  from this and carr
+      show "x \<in> 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 \<in> carrier R"
+  shows "(a_kernel R S h) +> a = {x \<in> 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 \<in> a_kernel R S h +> a"
+  from acarr and this
+      have xcarr: "x \<in> 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 \<in> {x \<in> 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 \<in> carrier R"
+     and hx: "h x = h a"
+  from acarr xcarr hx
+      show "x \<in> a_kernel R S h +> a" by (rule homeq_imp_rcos)
+qed
+
+end