src/HOL/Algebra/RingHom.thy
 author wenzelm Thu Jun 21 20:07:26 2007 +0200 (2007-06-21) changeset 23464 bc2563c37b1a parent 23463 9953ff53cc64 child 26204 da9778392d8c permissions -rw-r--r--
tuned proofs -- avoid implicit prems;
1 (*
2   Title:     HOL/Algebra/RingHom.thy
3   Id:        \$Id\$
4   Author:    Stephan Hohe, TU Muenchen
5 *)
7 theory RingHom
8 imports Ideal
9 begin
11 section {* Homomorphisms of Non-Commutative Rings *}
13 text {* Lifting existing lemmas in a @{text ring_hom_ring} locale *}
14 locale ring_hom_ring = ring R + ring S + var h +
15   assumes homh: "h \<in> ring_hom R S"
16   notes hom_mult [simp] = ring_hom_mult [OF homh]
17     and hom_one [simp] = ring_hom_one [OF homh]
19 interpretation ring_hom_cring \<subseteq> ring_hom_ring
20   by (unfold_locales, rule homh)
22 interpretation ring_hom_ring \<subseteq> abelian_group_hom R S
23 apply (rule abelian_group_homI)
24   apply (rule R.is_abelian_group)
25  apply (rule S.is_abelian_group)
26 apply (intro group_hom.intro group_hom_axioms.intro)
27   apply (rule R.a_group)
28  apply (rule S.a_group)
29 apply (insert homh, unfold hom_def ring_hom_def)
30 apply simp
31 done
33 lemma (in ring_hom_ring) is_ring_hom_ring:
34   includes struct R + struct S
35   shows "ring_hom_ring R S h"
36 by fact
38 lemma ring_hom_ringI:
39   includes ring R + ring S
40   assumes (* morphism: "h \<in> carrier R \<rightarrow> carrier S" *)
41           hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
42       and compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
43       and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
44       and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
45   shows "ring_hom_ring R S h"
46 apply unfold_locales
47 apply (unfold ring_hom_def, safe)
48    apply (simp add: hom_closed Pi_def)
49   apply (erule (1) compatible_mult)
50  apply (erule (1) compatible_add)
51 apply (rule compatible_one)
52 done
54 lemma ring_hom_ringI2:
55   includes ring R + ring S
56   assumes h: "h \<in> ring_hom R S"
57   shows "ring_hom_ring R S h"
58 apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro)
59 apply (rule R.is_ring)
60 apply (rule S.is_ring)
61 apply (rule h)
62 done
64 lemma ring_hom_ringI3:
65   includes abelian_group_hom R S + ring R + ring S
66   assumes compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
67       and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
68   shows "ring_hom_ring R S h"
69 apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro, rule R.is_ring, rule S.is_ring)
70 apply (insert group_hom.homh[OF a_group_hom])
71 apply (unfold hom_def ring_hom_def, simp)
72 apply safe
73 apply (erule (1) compatible_mult)
74 apply (rule compatible_one)
75 done
77 lemma ring_hom_cringI:
78   includes ring_hom_ring R S h + cring R + cring S
79   shows "ring_hom_cring R S h"
80   by (intro ring_hom_cring.intro ring_hom_cring_axioms.intro)
81     (rule R.is_cring, rule S.is_cring, rule homh)
84 subsection {* The kernel of a ring homomorphism *}
86 --"the kernel of a ring homomorphism is an ideal"
87 lemma (in ring_hom_ring) kernel_is_ideal:
88   shows "ideal (a_kernel R S h) R"
89 apply (rule idealI)
90    apply (rule R.is_ring)
92  apply (unfold a_kernel_def', simp+)
93 done
95 text {* Elements of the kernel are mapped to zero *}
96 lemma (in abelian_group_hom) kernel_zero [simp]:
97   "i \<in> a_kernel R S h \<Longrightarrow> h i = \<zero>\<^bsub>S\<^esub>"
98 by (simp add: a_kernel_defs)
101 subsection {* Cosets *}
103 text {* Cosets of the kernel correspond to the elements of the image of the homomorphism *}
104 lemma (in ring_hom_ring) rcos_imp_homeq:
105   assumes acarr: "a \<in> carrier R"
106       and xrcos: "x \<in> a_kernel R S h +> a"
107   shows "h x = h a"
108 proof -
109   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
111   from xrcos
112       have "\<exists>i \<in> a_kernel R S h. x = i \<oplus> a" by (simp add: a_r_coset_defs)
113   from this obtain i
114       where iker: "i \<in> a_kernel R S h"
115         and x: "x = i \<oplus> a"
116       by fast+
117   note carr = acarr iker[THEN a_Hcarr]
119   from x
120       have "h x = h (i \<oplus> a)" by simp
121   also from carr
122       have "\<dots> = h i \<oplus>\<^bsub>S\<^esub> h a" by simp
123   also from iker
124       have "\<dots> = \<zero>\<^bsub>S\<^esub> \<oplus>\<^bsub>S\<^esub> h a" by simp
125   also from carr
126       have "\<dots> = h a" by simp
127   finally
128       show "h x = h a" .
129 qed
131 lemma (in ring_hom_ring) homeq_imp_rcos:
132   assumes acarr: "a \<in> carrier R"
133       and xcarr: "x \<in> carrier R"
134       and hx: "h x = h a"
135   shows "x \<in> a_kernel R S h +> a"
136 proof -
137   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
139   note carr = acarr xcarr
140   note hcarr = acarr[THEN hom_closed] xcarr[THEN hom_closed]
142   from hx and hcarr
143       have a: "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = \<zero>\<^bsub>S\<^esub>" by algebra
144   from carr
145       have "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = h (x \<oplus> \<ominus>a)" by simp
146   from a and this
147       have b: "h (x \<oplus> \<ominus>a) = \<zero>\<^bsub>S\<^esub>" by simp
149   from carr have "x \<oplus> \<ominus>a \<in> carrier R" by simp
150   from this and b
151       have "x \<oplus> \<ominus>a \<in> a_kernel R S h"
152       unfolding a_kernel_def'
153       by fast
155   from this and carr
156       show "x \<in> a_kernel R S h +> a" by (simp add: a_rcos_module_rev)
157 qed
159 corollary (in ring_hom_ring) rcos_eq_homeq:
160   assumes acarr: "a \<in> carrier R"
161   shows "(a_kernel R S h) +> a = {x \<in> carrier R. h x = h a}"
162 apply rule defer 1
163 apply clarsimp defer 1
164 proof
165   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
167   fix x
168   assume xrcos: "x \<in> a_kernel R S h +> a"
169   from acarr and this
170       have xcarr: "x \<in> carrier R"
171       by (rule a_elemrcos_carrier)
173   from xrcos
174       have "h x = h a" by (rule rcos_imp_homeq[OF acarr])
175   from xcarr and this
176       show "x \<in> {x \<in> carrier R. h x = h a}" by fast
177 next
178   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
180   fix x
181   assume xcarr: "x \<in> carrier R"
182      and hx: "h x = h a"
183   from acarr xcarr hx
184       show "x \<in> a_kernel R S h +> a" by (rule homeq_imp_rcos)
185 qed
187 end