(* 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>S ⊕S 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
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 (x ⊗R y) = h x ⊗S h y;
!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> h (x ⊕R y) = h x ⊕S 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 (x ⊗R y) = h x ⊗S 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
lemma kernel_is_ideal:
ideal (a_kernel R S h) R
lemma kernel_zero:
i ∈ a_kernel R S h ==> h i = \<zero>S
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}