(* Title: HOL/Algebra/Ideal_Product.thy
Author: Paulo EmÃlio de Vilhena
*)
theory Ideal_Product
imports Ideal
begin
section \<open>Product of Ideals\<close>
text \<open>In this section, we study the structure of the set of ideals of a given ring.\<close>
inductive_set
ideal_prod :: "[ ('a, 'b) ring_scheme, 'a set, 'a set ] \<Rightarrow> 'a set" (infixl "\<cdot>\<index>" 80)
for R and I and J (* both I and J are supposed ideals *) where
prod: "\<lbrakk> i \<in> I; j \<in> J \<rbrakk> \<Longrightarrow> i \<otimes>\<^bsub>R\<^esub> j \<in> ideal_prod R I J"
| sum: "\<lbrakk> s1 \<in> ideal_prod R I J; s2 \<in> ideal_prod R I J \<rbrakk> \<Longrightarrow> s1 \<oplus>\<^bsub>R\<^esub> s2 \<in> ideal_prod R I J"
definition ideals_set :: "('a, 'b) ring_scheme \<Rightarrow> ('a set) ring"
where "ideals_set R = \<lparr> carrier = { I. ideal I R },
mult = ideal_prod R,
one = carrier R,
zero = { \<zero>\<^bsub>R\<^esub> },
add = set_add R \<rparr>"
subsection \<open>Basic Properties\<close>
lemma (in ring) ideal_prod_in_carrier:
assumes "ideal I R" "ideal J R"
shows "I \<cdot> J \<subseteq> carrier R"
proof
fix s assume "s \<in> I \<cdot> J" thus "s \<in> carrier R"
by (induct s rule: ideal_prod.induct) (auto, meson assms ideal.I_l_closed ideal.Icarr)
qed
lemma (in ring) ideal_prod_inter:
assumes "ideal I R" "ideal J R"
shows "I \<cdot> J \<subseteq> I \<inter> J"
proof
fix s assume "s \<in> I \<cdot> J" thus "s \<in> I \<inter> J"
apply (induct s rule: ideal_prod.induct)
apply (auto, (meson assms ideal.I_r_closed ideal.I_l_closed ideal.Icarr)+)
apply (simp_all add: additive_subgroup.a_closed assms ideal.axioms(1))
done
qed
lemma (in ring) ideal_prod_is_ideal:
assumes "ideal I R" "ideal J R"
shows "ideal (I \<cdot> J) R"
proof (rule idealI)
show "ring R" using ring_axioms .
next
show "subgroup (I \<cdot> J) (add_monoid R)"
unfolding subgroup_def
proof (auto)
show "\<zero> \<in> I \<cdot> J" using ideal_prod.prod[of \<zero> I \<zero> J R]
by (simp add: additive_subgroup.zero_closed assms ideal.axioms(1))
next
fix s1 s2 assume s1: "s1 \<in> I \<cdot> J" and s2: "s2 \<in> I \<cdot> J"
have IJcarr: "\<And>a. a \<in> I \<cdot> J \<Longrightarrow> a \<in> carrier R"
by (meson assms subsetD ideal_prod_in_carrier)
show "s1 \<in> carrier R" using ideal_prod_in_carrier[OF assms] s1 by blast
show "s1 \<oplus> s2 \<in> I \<cdot> J" by (simp add: ideal_prod.sum[OF s1 s2])
show "inv\<^bsub>add_monoid R\<^esub> s1 \<in> I \<cdot> J" using s1
proof (induct s1 rule: ideal_prod.induct)
case (prod i j)
hence "inv\<^bsub>add_monoid R\<^esub> (i \<otimes> j) = (inv\<^bsub>add_monoid R\<^esub> i) \<otimes> j"
by (metis a_inv_def assms(1) assms(2) ideal.Icarr l_minus)
thus ?case using ideal_prod.prod[of "inv\<^bsub>add_monoid R\<^esub> i" I j J R] assms
by (simp add: additive_subgroup.a_subgroup ideal.axioms(1) prod.hyps subgroup.m_inv_closed)
next
case (sum s1 s2) thus ?case
by (metis (no_types) IJcarr a_inv_def add.inv_mult_group ideal_prod.sum sum.hyps)
qed
qed
next
fix s x assume s: "s \<in> I \<cdot> J" and x: "x \<in> carrier R"
show "x \<otimes> s \<in> I \<cdot> J" using s
proof (induct s rule: ideal_prod.induct)
case (prod i j) thus ?case using ideal_prod.prod[of "x \<otimes> i" I j J R] assms
by (simp add: x ideal.I_l_closed ideal.Icarr m_assoc)
next
case (sum s1 s2) thus ?case
proof -
have IJ: "I \<cdot> J \<subseteq> carrier R"
by (metis (no_types) assms(1) assms(2) ideal.axioms(2) ring.ideal_prod_in_carrier)
then have "s2 \<in> carrier R"
using sum.hyps(3) by blast
moreover have "s1 \<in> carrier R"
using IJ sum.hyps(1) by blast
ultimately show ?thesis
by (simp add: ideal_prod.sum r_distr sum.hyps x)
qed
qed
show "s \<otimes> x \<in> I \<cdot> J" using s
proof (induct s rule: ideal_prod.induct)
case (prod i j) thus ?case using ideal_prod.prod[of i I "j \<otimes> x" J R] assms x
by (simp add: x ideal.I_r_closed ideal.Icarr m_assoc)
next
case (sum s1 s2) thus ?case
proof -
have "s1 \<in> carrier R" "s2 \<in> carrier R"
by (meson assms subsetD ideal_prod_in_carrier sum.hyps)+
then show ?thesis
by (metis ideal_prod.sum l_distr sum.hyps(2) sum.hyps(4) x)
qed
qed
qed
lemma (in ring) ideal_prod_eq_genideal:
assumes "ideal I R" "ideal J R"
shows "I \<cdot> J = Idl (I <#> J)"
proof
have "I <#> J \<subseteq> I \<cdot> J"
proof
fix s assume "s \<in> I <#> J"
then obtain i j where "i \<in> I" "j \<in> J" "s = i \<otimes> j"
unfolding set_mult_def by blast
thus "s \<in> I \<cdot> J" using ideal_prod.prod by simp
qed
thus "Idl (I <#> J) \<subseteq> I \<cdot> J"
unfolding genideal_def using ideal_prod_is_ideal[OF assms] by blast
next
show "I \<cdot> J \<subseteq> Idl (I <#> J)"
proof
fix s assume "s \<in> I \<cdot> J" thus "s \<in> Idl (I <#> J)"
proof (induct s rule: ideal_prod.induct)
case (prod i j) hence "i \<otimes> j \<in> I <#> J" unfolding set_mult_def by blast
thus ?case unfolding genideal_def by blast
next
case (sum s1 s2) thus ?case
by (simp add: additive_subgroup.a_closed additive_subgroup.a_subset
assms genideal_ideal ideal.axioms(1) set_mult_closed)
qed
qed
qed
lemma (in ring) ideal_prod_simp:
assumes "ideal I R" "ideal J R" (* the second assumption could be suppressed *)
shows "I = I <+> (I \<cdot> J)"
proof
show "I \<subseteq> I <+> I \<cdot> J"
proof
fix i assume "i \<in> I" hence "i \<oplus> \<zero> \<in> I <+> I \<cdot> J"
using set_add_def'[of R I "I \<cdot> J"] ideal_prod_is_ideal[OF assms]
additive_subgroup.zero_closed[OF ideal.axioms(1), of "I \<cdot> J" R] by auto
thus "i \<in> I <+> I \<cdot> J"
using \<open>i \<in> I\<close> assms(1) ideal.Icarr by fastforce
qed
next
show "I <+> I \<cdot> J \<subseteq> I"
proof
fix s assume "s \<in> I <+> I \<cdot> J"
then obtain i ij where "i \<in> I" "ij \<in> I \<cdot> J" "s = i \<oplus> ij"
using set_add_def'[of R I "I \<cdot> J"] by auto
thus "s \<in> I"
using ideal_prod_inter[OF assms]
by (meson additive_subgroup.a_closed assms(1) ideal.axioms(1) inf_sup_ord(1) subsetCE)
qed
qed
lemma (in ring) ideal_prod_one:
assumes "ideal I R"
shows "I \<cdot> (carrier R) = I"
proof
show "I \<cdot> (carrier R) \<subseteq> I"
proof
fix s assume "s \<in> I \<cdot> (carrier R)" thus "s \<in> I"
by (induct s rule: ideal_prod.induct)
(simp_all add: assms ideal.I_r_closed additive_subgroup.a_closed ideal.axioms(1))
qed
next
show "I \<subseteq> I \<cdot> (carrier R)"
proof
fix i assume "i \<in> I" thus "i \<in> I \<cdot> (carrier R)"
by (metis assms ideal.Icarr ideal_prod.simps one_closed r_one)
qed
qed
lemma (in ring) ideal_prod_zero:
assumes "ideal I R"
shows "I \<cdot> { \<zero> } = { \<zero> }"
proof
show "I \<cdot> { \<zero> } \<subseteq> { \<zero> }"
proof
fix s assume "s \<in> I \<cdot> {\<zero>}" thus "s \<in> { \<zero> }"
using assms ideal.Icarr by (induct s rule: ideal_prod.induct) (fastforce, simp)
qed
next
show "{ \<zero> } \<subseteq> I \<cdot> { \<zero> }"
by (simp add: additive_subgroup.zero_closed assms
ideal.axioms(1) ideal_prod_is_ideal zeroideal)
qed
lemma (in ring) ideal_prod_assoc:
assumes "ideal I R" "ideal J R" "ideal K R"
shows "(I \<cdot> J) \<cdot> K = I \<cdot> (J \<cdot> K)"
proof
show "(I \<cdot> J) \<cdot> K \<subseteq> I \<cdot> (J \<cdot> K)"
proof
fix s assume "s \<in> (I \<cdot> J) \<cdot> K" thus "s \<in> I \<cdot> (J \<cdot> K)"
proof (induct s rule: ideal_prod.induct)
case (sum s1 s2) thus ?case
by (simp add: ideal_prod.sum)
next
case (prod i k) thus ?case
proof (induct i rule: ideal_prod.induct)
case (prod i j) thus ?case
using ideal_prod.prod[OF prod(1) ideal_prod.prod[OF prod(2-3),of R], of R]
by (metis assms ideal.Icarr m_assoc)
next
case (sum s1 s2) thus ?case
proof -
have "s1 \<in> carrier R" "s2 \<in> carrier R"
by (meson assms subsetD ideal.axioms(2) ring.ideal_prod_in_carrier sum.hyps)+
moreover have "k \<in> carrier R"
by (meson additive_subgroup.a_Hcarr assms(3) ideal.axioms(1) sum.prems)
ultimately show ?thesis
by (metis ideal_prod.sum l_distr sum.hyps(2) sum.hyps(4) sum.prems)
qed
qed
qed
qed
next
show "I \<cdot> (J \<cdot> K) \<subseteq> (I \<cdot> J) \<cdot> K"
proof
fix s assume "s \<in> I \<cdot> (J \<cdot> K)" thus "s \<in> (I \<cdot> J) \<cdot> K"
proof (induct s rule: ideal_prod.induct)
case (sum s1 s2) thus ?case by (simp add: ideal_prod.sum)
next
case (prod i j) show ?case using prod(2) prod(1)
proof (induct j rule: ideal_prod.induct)
case (prod j k) thus ?case
using ideal_prod.prod[OF ideal_prod.prod[OF prod(3) prod(1), of R] prod (2), of R]
by (metis assms ideal.Icarr m_assoc)
next
case (sum s1 s2) thus ?case
proof -
have "\<And>a A B. \<lbrakk>a \<in> B \<cdot> A; ideal A R; ideal B R\<rbrakk> \<Longrightarrow> a \<in> carrier R"
by (meson subsetD ideal_prod_in_carrier)
moreover have "i \<in> carrier R"
by (meson additive_subgroup.a_Hcarr assms(1) ideal.axioms(1) sum.prems)
ultimately show ?thesis
by (metis (no_types) assms(2) assms(3) ideal_prod.sum r_distr sum)
qed
qed
qed
qed
qed
lemma (in ring) ideal_prod_r_distr:
assumes "ideal I R" "ideal J R" "ideal K R"
shows "I \<cdot> (J <+> K) = (I \<cdot> J) <+> (I \<cdot> K)"
proof
show "I \<cdot> (J <+> K) \<subseteq> I \<cdot> J <+> I \<cdot> K"
proof
fix s assume "s \<in> I \<cdot> (J <+> K)" thus "s \<in> I \<cdot> J <+> I \<cdot> K"
proof(induct s rule: ideal_prod.induct)
case (prod i jk)
then obtain j k where j: "j \<in> J" and k: "k \<in> K" and jk: "jk = j \<oplus> k"
using set_add_def'[of R J K] by auto
hence "i \<otimes> j \<oplus> i \<otimes> k \<in> I \<cdot> J <+> I \<cdot> K"
using ideal_prod.prod[OF prod(1) j,of R]
ideal_prod.prod[OF prod(1) k,of R]
set_add_def'[of R "I \<cdot> J" "I \<cdot> K"] by auto
thus ?case
using assms ideal.Icarr r_distr jk j k prod(1) by metis
next
case (sum s1 s2) thus ?case
by (simp add: add_ideals additive_subgroup.a_closed assms ideal.axioms(1)
local.ring_axioms ring.ideal_prod_is_ideal)
qed
qed
next
{ fix s J K assume A: "ideal J R" "ideal K R" "s \<in> I \<cdot> J"
have "s \<in> I \<cdot> (J <+> K) \<and> s \<in> I \<cdot> (K <+> J)"
proof -
from \<open>s \<in> I \<cdot> J\<close> have "s \<in> I \<cdot> (J <+> K)"
proof (induct s rule: ideal_prod.induct)
case (prod i j)
hence "(j \<oplus> \<zero>) \<in> J <+> K"
using set_add_def'[of R J K]
additive_subgroup.zero_closed[OF ideal.axioms(1), of K R] A(2) by auto
thus ?case
by (metis A(1) additive_subgroup.a_Hcarr ideal.axioms(1) ideal_prod.prod prod r_zero)
next
case (sum s1 s2) thus ?case
by (simp add: ideal_prod.sum)
qed
thus ?thesis
by (metis A(1) A(2) ideal_def ring.union_genideal sup_commute)
qed } note aux_lemma = this
show "I \<cdot> J <+> I \<cdot> K \<subseteq> I \<cdot> (J <+> K)"
proof
fix s assume "s \<in> I \<cdot> J <+> I \<cdot> K"
then obtain s1 s2 where s1: "s1 \<in> I \<cdot> J" and s2: "s2 \<in> I \<cdot> K" and s: "s = s1 \<oplus> s2"
using set_add_def'[of R "I \<cdot> J" "I \<cdot> K"] by auto
thus "s \<in> I \<cdot> (J <+> K)"
using aux_lemma[OF assms(2) assms(3) s1]
aux_lemma[OF assms(3) assms(2) s2] by (simp add: ideal_prod.sum)
qed
qed
lemma (in cring) ideal_prod_commute:
assumes "ideal I R" "ideal J R"
shows "I \<cdot> J = J \<cdot> I"
proof -
{ fix I J assume A: "ideal I R" "ideal J R"
have "I \<cdot> J \<subseteq> J \<cdot> I"
proof
fix s assume "s \<in> I \<cdot> J" thus "s \<in> J \<cdot> I"
proof (induct s rule: ideal_prod.induct)
case (prod i j) thus ?case
using m_comm[OF ideal.Icarr[OF A(1) prod(1)] ideal.Icarr[OF A(2) prod(2)]]
by (simp add: ideal_prod.prod)
next
case (sum s1 s2) thus ?case by (simp add: ideal_prod.sum)
qed
qed }
thus ?thesis using assms by blast
qed
text \<open>The following result would also be true for locale ring\<close>
lemma (in cring) ideal_prod_distr:
assumes "ideal I R" "ideal J R" "ideal K R"
shows "I \<cdot> (J <+> K) = (I \<cdot> J) <+> (I \<cdot> K)"
and "(J <+> K) \<cdot> I = (J \<cdot> I) <+> (K \<cdot> I)"
by (simp_all add: assms ideal_prod_commute local.ring_axioms
ring.add_ideals ring.ideal_prod_r_distr)
lemma (in cring) ideal_prod_eq_inter:
assumes "ideal I R" "ideal J R"
and "I <+> J = carrier R"
shows "I \<cdot> J = I \<inter> J"
proof
show "I \<cdot> J \<subseteq> I \<inter> J"
using assms ideal_prod_inter by auto
next
show "I \<inter> J \<subseteq> I \<cdot> J"
proof
have "\<one> \<in> I <+> J" using assms(3) one_closed by simp
then obtain i j where ij: "i \<in> I" "j \<in> J" "\<one> = i \<oplus> j"
using set_add_def'[of R I J] by auto
fix s assume s: "s \<in> I \<inter> J"
hence "(i \<otimes> s \<in> I \<cdot> J) \<and> (s \<otimes> j \<in> I \<cdot> J)"
using ij(1-2) by (simp add: ideal_prod.prod)
moreover have "s = (i \<otimes> s) \<oplus> (s \<otimes> j)"
using ideal.Icarr[OF assms(1) ij(1)]
ideal.Icarr[OF assms(2) ij(2)]
ideal.Icarr[OF assms(1), of s]
by (metis ij(3) s m_comm[of s i] Int_iff r_distr r_one)
ultimately show "s \<in> I \<cdot> J"
using ideal_prod.sum by fastforce
qed
qed
subsection \<open>Structure of the Set of Ideals\<close>
text \<open>We focus on commutative rings for convenience.\<close>
lemma (in cring) ideals_set_is_semiring: "semiring (ideals_set R)"
proof -
have "abelian_monoid (ideals_set R)"
apply (rule abelian_monoidI) unfolding ideals_set_def
apply (simp_all add: add_ideals zeroideal)
apply (simp add: add.set_mult_assoc additive_subgroup.a_subset ideal.axioms(1) set_add_defs(1))
apply (metis Un_absorb1 additive_subgroup.a_subset additive_subgroup.zero_closed
cgenideal_minimal cgenideal_self empty_iff genideal_minimal ideal.axioms(1)
local.ring_axioms order_refl ring.genideal_self subset_antisym subset_singletonD
union_genideal zero_closed zeroideal)
by (metis sup_commute union_genideal)
moreover have "monoid (ideals_set R)"
apply (rule monoidI) unfolding ideals_set_def
apply (simp_all add: ideal_prod_is_ideal oneideal
ideal_prod_commute ideal_prod_one)
by (metis ideal_prod_assoc ideal_prod_commute)
ultimately show ?thesis
unfolding semiring_def semiring_axioms_def ideals_set_def
by (simp_all add: ideal_prod_distr ideal_prod_commute ideal_prod_zero zeroideal)
qed
lemma (in cring) ideals_set_is_comm_monoid: "comm_monoid (ideals_set R)"
proof -
have "monoid (ideals_set R)"
apply (rule monoidI) unfolding ideals_set_def
apply (simp_all add: ideal_prod_is_ideal oneideal
ideal_prod_commute ideal_prod_one)
by (metis ideal_prod_assoc ideal_prod_commute)
thus ?thesis
unfolding comm_monoid_def comm_monoid_axioms_def
by (simp add: ideal_prod_commute ideals_set_def)
qed
lemma (in cring) ideal_prod_eq_Inter_aux:
assumes "I: {..(Suc n)} \<rightarrow> { J. ideal J R }"
and "\<And>i j. \<lbrakk> i \<le> Suc n; j \<le> Suc n \<rbrakk> \<Longrightarrow>
i \<noteq> j \<Longrightarrow> (I i) <+> (I j) = carrier R"
shows "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..n}. I k) <+> (I (Suc n)) = carrier R" using assms
proof (induct n arbitrary: I)
case 0
hence "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..0}. I k) <+> I (Suc 0) = (I 0) <+> (I (Suc 0))"
using comm_monoid.finprod_0[OF ideals_set_is_comm_monoid, of I]
by (simp add: atMost_Suc ideals_set_def)
also have " ... = carrier R"
using 0(2)[of 0 "Suc 0"] by simp
finally show ?case .
next
interpret ISet: comm_monoid "ideals_set R"
by (simp add: ideals_set_is_comm_monoid)
case (Suc n)
let ?I' = "\<lambda>i. I (Suc i)"
have "?I': {..(Suc n)} \<rightarrow> { J. ideal J R }"
using Suc.prems(1) by auto
moreover have "\<And>i j. \<lbrakk> i \<le> Suc n; j \<le> Suc n \<rbrakk> \<Longrightarrow>
i \<noteq> j \<Longrightarrow> (?I' i) <+> (?I' j) = carrier R"
by (simp add: Suc.prems(2))
ultimately have "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..n}. ?I' k) <+> (?I' (Suc n)) = carrier R"
using Suc.hyps by metis
moreover have I_carr: "I: {..Suc (Suc n)} \<rightarrow> carrier (ideals_set R)"
unfolding ideals_set_def using Suc by simp
hence I'_carr: "I \<in> Suc ` {..n} \<rightarrow> carrier (ideals_set R)" by auto
ultimately have "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {(Suc 0)..Suc n}. I k) <+> (I (Suc (Suc n))) = carrier R"
using ISet.finprod_reindex[of I "\<lambda>i. Suc i" "{..n}"] by (simp add: atMost_atLeast0)
hence "(carrier R) \<cdot> (I 0) = ((\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {Suc 0..Suc n}. I k) <+> I (Suc (Suc n))) \<cdot> (I 0)"
by auto
moreover have fprod_cl1: "ideal (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {Suc 0..Suc n}. I k) R"
by (metis I'_carr ISet.finprod_closed One_nat_def ideals_set_def image_Suc_atMost
mem_Collect_eq partial_object.select_convs(1))
ultimately
have "I 0 = (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {Suc 0..Suc n}. I k) \<cdot> (I 0) <+> I (Suc (Suc n)) \<cdot> (I 0)"
by (metis PiE Suc.prems(1) atLeast0_atMost_Suc atLeast0_atMost_Suc_eq_insert_0
atMost_atLeast0 ideal_prod_commute ideal_prod_distr(2) ideal_prod_one insertI1
mem_Collect_eq oneideal)
also have " ... = (I 0) \<cdot> (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {Suc 0..Suc n}. I k) <+> I (Suc (Suc n)) \<cdot> (I 0)"
using fprod_cl1 ideal_prod_commute Suc.prems(1)
by (simp add: atLeast0_atMost_Suc_eq_insert_0 atMost_atLeast0)
also have " ... = (I 0) \<otimes>\<^bsub>(ideals_set R)\<^esub> (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {Suc 0..Suc n}. I k) <+>
I (Suc (Suc n)) \<cdot> (I 0)"
by (simp add: ideals_set_def)
finally have I0: "I 0 = (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) <+> I (Suc (Suc n)) \<cdot> (I 0)"
using ISet.finprod_insert[of "{Suc 0..Suc n}" 0 I]
I_carr I'_carr atMost_atLeast0 ISet.finprod_0' atMost_Suc by auto
have I_SucSuc_I0: "ideal (I (Suc (Suc n))) R \<and> ideal (I 0) R"
using Suc.prems(1) by auto
have fprod_cl2: "ideal (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) R"
by (metis (no_types) ISet.finprod_closed I_carr Pi_split_insert_domain atMost_Suc ideals_set_def mem_Collect_eq partial_object.select_convs(1))
have "carrier R = I (Suc (Suc n)) <+> I 0"
by (simp add: Suc.prems(2))
also have " ... = I (Suc (Suc n)) <+>
((\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) <+> I (Suc (Suc n)) \<cdot> (I 0))"
using I0 by auto
also have " ... = I (Suc (Suc n)) <+>
(I (Suc (Suc n)) \<cdot> (I 0) <+> (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k))"
using fprod_cl2 I_SucSuc_I0 by (metis Un_commute ideal_prod_is_ideal union_genideal)
also have " ... = (I (Suc (Suc n)) <+> I (Suc (Suc n)) \<cdot> (I 0)) <+>
(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k)"
using fprod_cl2 I_SucSuc_I0 by (metis add.set_mult_assoc ideal_def ideal_prod_in_carrier
oneideal ring.ideal_prod_one set_add_defs(1))
also have " ... = I (Suc (Suc n)) <+> (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k)"
using ideal_prod_simp[of "I (Suc (Suc n))" "I 0"] I_SucSuc_I0 by simp
also have " ... = (\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) <+> I (Suc (Suc n))"
using fprod_cl2 I_SucSuc_I0 by (metis Un_commute union_genideal)
finally show ?case by simp
qed
theorem (in cring) ideal_prod_eq_Inter:
assumes "I: {..n :: nat} \<rightarrow> { J. ideal J R }"
and "\<And>i j. \<lbrakk> i \<in> {..n}; j \<in> {..n} \<rbrakk> \<Longrightarrow> i \<noteq> j \<Longrightarrow> (I i) <+> (I j) = carrier R"
shows "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..n}. I k) = (\<Inter> k \<in> {..n}. I k)" using assms
proof (induct n)
case 0 thus ?case
using comm_monoid.finprod_0[OF ideals_set_is_comm_monoid] by (simp add: ideals_set_def)
next
interpret ISet: comm_monoid "ideals_set R"
by (simp add: ideals_set_is_comm_monoid)
case (Suc n)
hence IH: "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..n}. I k) = (\<Inter> k \<in> {..n}. I k)"
by (simp add: atMost_Suc)
hence "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) = I (Suc n) \<otimes>\<^bsub>(ideals_set R)\<^esub> (\<Inter> k \<in> {..n}. I k)"
using ISet.finprod_insert[of "{Suc 0..Suc n}" 0 I] atMost_Suc_eq_insert_0[of n]
by (metis ISet.finprod_Suc Suc.prems(1) ideals_set_def partial_object.select_convs(1))
hence "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) = I (Suc n) \<cdot> (\<Inter> k \<in> {..n}. I k)"
by (simp add: ideals_set_def)
moreover have "(\<Inter> k \<in> {..n}. I k) <+> I (Suc n) = carrier R"
using ideal_prod_eq_Inter_aux[of I n] by (simp add: Suc.prems IH)
moreover have "ideal (\<Inter> k \<in> {..n}. I k) R"
using ring.i_Intersect[of R "I ` {..n}"]
by (metis IH ISet.finprod_closed Pi_split_insert_domain Suc.prems(1) atMost_Suc
ideals_set_def mem_Collect_eq partial_object.select_convs(1))
ultimately
have "(\<Otimes>\<^bsub>(ideals_set R)\<^esub> k \<in> {..Suc n}. I k) = (\<Inter> k \<in> {..n}. I k) \<inter> I (Suc n)"
using ideal_prod_eq_inter[of "\<Inter> k \<in> {..n}. I k" "I (Suc n)"]
ideal_prod_commute[of "\<Inter> k \<in> {..n}. I k" "I (Suc n)"]
by (metis PiE Suc.prems(1) atMost_iff mem_Collect_eq order_refl)
thus ?case by (simp add: Int_commute atMost_Suc)
qed
corollary (in cring) inter_plus_ideal_eq_carrier:
assumes "\<And>i. i \<le> Suc n \<Longrightarrow> ideal (I i) R"
and "\<And>i j. \<lbrakk> i \<le> Suc n; j \<le> Suc n; i \<noteq> j \<rbrakk> \<Longrightarrow> I i <+> I j = carrier R"
shows "(\<Inter> i \<le> n. I i) <+> (I (Suc n)) = carrier R"
using ideal_prod_eq_Inter[of I n] ideal_prod_eq_Inter_aux[of I n] by (auto simp add: assms)
corollary (in cring) inter_plus_ideal_eq_carrier_arbitrary:
assumes "\<And>i. i \<le> Suc n \<Longrightarrow> ideal (I i) R"
and "\<And>i j. \<lbrakk> i \<le> Suc n; j \<le> Suc n; i \<noteq> j \<rbrakk> \<Longrightarrow> I i <+> I j = carrier R"
and "j \<le> Suc n"
shows "(\<Inter> i \<in> ({..(Suc n)} - { j }). I i) <+> (I j) = carrier R"
proof -
define I' where "I' = (\<lambda>i. if i = Suc n then (I j) else
if i = j then (I (Suc n))
else (I i))"
have "\<And>i. i \<le> Suc n \<Longrightarrow> ideal (I' i) R"
using I'_def assms(1) assms(3) by auto
moreover have "\<And>i j. \<lbrakk> i \<le> Suc n; j \<le> Suc n; i \<noteq> j \<rbrakk> \<Longrightarrow> I' i <+> I' j = carrier R"
using I'_def assms(2-3) by force
ultimately have "(\<Inter> i \<le> n. I' i) <+> (I' (Suc n)) = carrier R"
using inter_plus_ideal_eq_carrier by simp
moreover have "I' ` {..n} = I ` ({..(Suc n)} - { j })"
proof
show "I' ` {..n} \<subseteq> I ` ({..Suc n} - {j})"
proof
fix x assume "x \<in> I' ` {..n}"
then obtain i where i: "i \<in> {..n}" "I' i = x" by blast
thus "x \<in> I ` ({..Suc n} - {j})"
proof (cases)
assume "i = j" thus ?thesis using i I'_def by auto
next
assume "i \<noteq> j" thus ?thesis using I'_def i insert_iff by auto
qed
qed
next
show "I ` ({..Suc n} - {j}) \<subseteq> I' ` {..n}"
proof
fix x assume "x \<in> I ` ({..Suc n} - {j})"
then obtain i where i: "i \<in> {..Suc n}" "i \<noteq> j" "I i = x" by blast
thus "x \<in> I' ` {..n}"
proof (cases)
assume "i = Suc n" thus ?thesis using I'_def assms(3) i(2-3) by auto
next
assume "i \<noteq> Suc n" thus ?thesis using I'_def i by auto
qed
qed
qed
ultimately show ?thesis using I'_def by metis
qed
subsection \<open>Another Characterization of Prime Ideals\<close>
text \<open>With product of ideals being defined, we can give another definition of a prime ideal\<close>
lemma (in ring) primeideal_divides_ideal_prod:
assumes "primeideal P R" "ideal I R" "ideal J R"
and "I \<cdot> J \<subseteq> P"
shows "I \<subseteq> P \<or> J \<subseteq> P"
proof (cases)
assume "\<exists> i \<in> I. i \<notin> P"
then obtain i where i: "i \<in> I" "i \<notin> P" by blast
have "J \<subseteq> P"
proof
fix j assume j: "j \<in> J"
hence "i \<otimes> j \<in> P"
using ideal_prod.prod[OF i(1) j, of R] assms(4) by auto
thus "j \<in> P"
using primeideal.I_prime[OF assms(1), of i j] i j
by (meson assms(2-3) ideal.Icarr)
qed
thus ?thesis by blast
next
assume "\<not> (\<exists> i \<in> I. i \<notin> P)" thus ?thesis by blast
qed
lemma (in cring) divides_ideal_prod_imp_primeideal:
assumes "ideal P R"
and "P \<noteq> carrier R"
and "\<And>I J. \<lbrakk> ideal I R; ideal J R; I \<cdot> J \<subseteq> P \<rbrakk> \<Longrightarrow> I \<subseteq> P \<or> J \<subseteq> P"
shows "primeideal P R"
proof -
have "\<And>a b. \<lbrakk> a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> P \<rbrakk> \<Longrightarrow> a \<in> P \<or> b \<in> P"
proof -
fix a b assume A: "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b \<in> P"
have "(PIdl a) \<cdot> (PIdl b) = Idl (PIdl (a \<otimes> b))"
using ideal_prod_eq_genideal[of "Idl { a }" "Idl { b }"]
A(1-2) cgenideal_eq_genideal cgenideal_ideal cgenideal_prod by auto
hence "(PIdl a) \<cdot> (PIdl b) = PIdl (a \<otimes> b)"
by (simp add: A Idl_subset_ideal cgenideal_ideal cgenideal_minimal
genideal_self oneideal subset_antisym)
hence "(PIdl a) \<cdot> (PIdl b) \<subseteq> P"
by (simp add: A(3) assms(1) cgenideal_minimal)
hence "(PIdl a) \<subseteq> P \<or> (PIdl b) \<subseteq> P"
by (simp add: A assms(3) cgenideal_ideal)
thus "a \<in> P \<or> b \<in> P"
using A cgenideal_self by blast
qed
thus ?thesis
using assms is_cring by (simp add: primeidealI)
qed
end