src/HOL/Algebra/RingHom.thy
 author wenzelm Thu, 23 Nov 2006 20:34:21 +0100 changeset 21502 7f3ea2b3bab6 parent 20318 0e0ea63fe768 child 23350 50c5b0912a0c permissions -rw-r--r--
prefer antiquotations over LaTeX macros;
```
(*
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 @{text 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 (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>"

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
```