(* 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 \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 standard (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 \\<^bsub>S\<^esub> h y"
and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \ y) = h x \\<^bsub>S\<^esub> h y"
and compatible_one: "h \ = \\<^bsub>S\<^esub>"
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 \\<^bsub>S\<^esub> h y"
and compatible_one: "h \ = \\<^bsub>S\<^esub>"
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 = \\<^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 \ 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 \\<^bsub>S\<^esub> h a" by simp
also from iker
have "\ = \\<^bsub>S\<^esub> \\<^bsub>S\<^esub> 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 \\<^bsub>S\<^esub> \\<^bsub>S\<^esub>h a = \\<^bsub>S\<^esub>" by algebra
from carr
have "h x \\<^bsub>S\<^esub> \\<^bsub>S\<^esub>h a = h (x \ \a)" by simp
from a and this
have b: "h (x \ \a) = \\<^bsub>S\<^esub>" by simp
from carr have "x \ \a \ carrier R" by simp
from this and b
have "x \ \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