(*

  Title:     HOL/Algebra/CIdeal.thy

  Id:        $Id$

  Author:    Stephan Hohe, TU Muenchen

*)

theory Ideal

imports Ring AbelCoset

begin

section {* Ideals *}

subsection {* General definition *}

locale ideal = additive_subgroup I R + ring R +

  assumes I_l_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"

      and I_r_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"

interpretation ideal \<subseteq> abelian_subgroup I R

apply (intro abelian_subgroupI3 abelian_group.intro)

  apply (rule ideal.axioms, rule ideal_axioms)

 apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)

apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)

done

lemma (in ideal) is_ideal:

  "ideal I R"

by (rule ideal_axioms)

lemma idealI:

  fixes R (structure)

  assumes "ring R"

  assumes a_subgroup: "subgroup I \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"

      and I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"

      and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"

  shows "ideal I R"

proof -

  interpret ring [R] by fact

  show ?thesis  apply (intro ideal.intro ideal_axioms.intro additive_subgroupI)

     apply (rule a_subgroup)

    apply (rule is_ring)

   apply (erule (1) I_l_closed)

  apply (erule (1) I_r_closed)

  done

qed

subsection {* Ideals Generated by a Subset of @{term [locale=ring] "carrier R"} *}

constdefs (structure R)

  genideal :: "('a, 'b) ring_scheme \<Rightarrow> 'a set \<Rightarrow> 'a set"  ("Idl\<index> _" [80] 79)

  "genideal R S \<equiv> Inter {I. ideal I R \<and> S \<subseteq> I}"

subsection {* Principal Ideals *}

locale principalideal = ideal +

  assumes generate: "\<exists>i \<in> carrier R. I = Idl {i}"

lemma (in principalideal) is_principalideal:

  shows "principalideal I R"

by (rule principalideal_axioms)

lemma principalidealI:

  fixes R (structure)

  assumes "ideal I R"

  assumes generate: "\<exists>i \<in> carrier R. I = Idl {i}"

  shows "principalideal I R"

proof -

  interpret ideal [I R] by fact

  show ?thesis  by (intro principalideal.intro principalideal_axioms.intro) (rule is_ideal, rule generate)

qed

subsection {* Maximal Ideals *}

locale maximalideal = ideal +

  assumes I_notcarr: "carrier R \<noteq> I"

      and I_maximal: "\<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"

lemma (in maximalideal) is_maximalideal:

 shows "maximalideal I R"

by (rule maximalideal_axioms)

lemma maximalidealI:

  fixes R

  assumes "ideal I R"

  assumes I_notcarr: "carrier R \<noteq> I"

     and I_maximal: "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"

  shows "maximalideal I R"

proof -

  interpret ideal [I R] by fact

  show ?thesis by (intro maximalideal.intro maximalideal_axioms.intro)

    (rule is_ideal, rule I_notcarr, rule I_maximal)

qed

subsection {* Prime Ideals *}

locale primeideal = ideal + cring +

  assumes I_notcarr: "carrier R \<noteq> I"

      and I_prime: "\<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"

lemma (in primeideal) is_primeideal:

 shows "primeideal I R"

by (rule primeideal_axioms)

lemma primeidealI:

  fixes R (structure)

  assumes "ideal I R"

  assumes "cring R"

  assumes I_notcarr: "carrier R \<noteq> I"

      and I_prime: "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"

  shows "primeideal I R"

proof -

  interpret ideal [I R] by fact

  interpret cring [R] by fact

  show ?thesis by (intro primeideal.intro primeideal_axioms.intro)

    (rule is_ideal, rule is_cring, rule I_notcarr, rule I_prime)

qed

lemma primeidealI2:

  fixes R (structure)

  assumes "additive_subgroup I R"

  assumes "cring R"

  assumes I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"

      and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"

      and I_notcarr: "carrier R \<noteq> I"

      and I_prime: "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"

  shows "primeideal I R"

proof -

  interpret additive_subgroup [I R] by fact

  interpret cring [R] by fact

  show ?thesis apply (intro_locales)

    apply (intro ideal_axioms.intro)

    apply (erule (1) I_l_closed)

    apply (erule (1) I_r_closed)

    apply (intro primeideal_axioms.intro)

    apply (rule I_notcarr)

    apply (erule (2) I_prime)

    done

qed

section {* Properties of Ideals *}

subsection {* Special Ideals *}

lemma (in ring) zeroideal:

  shows "ideal {\<zero>} R"

apply (intro idealI subgroup.intro)

      apply (rule is_ring)

     apply simp+

  apply (fold a_inv_def, simp)

 apply simp+

done

lemma (in ring) oneideal:

  shows "ideal (carrier R) R"

apply (intro idealI  subgroup.intro)

      apply (rule is_ring)

     apply simp+

  apply (fold a_inv_def, simp)

 apply simp+

done

lemma (in "domain") zeroprimeideal:

 shows "primeideal {\<zero>} R"

apply (intro primeidealI)

   apply (rule zeroideal)

  apply (rule domain.axioms, rule domain_axioms)

 defer 1

 apply (simp add: integral)

proof (rule ccontr, simp)

  assume "carrier R = {\<zero>}"

  from this have "\<one> = \<zero>" by (rule one_zeroI)

  from this and one_not_zero

      show "False" by simp

qed

subsection {* General Ideal Properies *}

lemma (in ideal) one_imp_carrier:

  assumes I_one_closed: "\<one> \<in> I"

  shows "I = carrier R"

apply (rule)

apply (rule)

apply (rule a_Hcarr, simp)

proof

  fix x

  assume xcarr: "x \<in> carrier R"

  from I_one_closed and this

      have "x \<otimes> \<one> \<in> I" by (intro I_l_closed)

  from this and xcarr

      show "x \<in> I" by simp

qed

lemma (in ideal) Icarr:

  assumes iI: "i \<in> I"

  shows "i \<in> carrier R"

using iI by (rule a_Hcarr)

subsection {* Intersection of Ideals *}

text {* \paragraph{Intersection of two ideals} The intersection of any

  two ideals is again an ideal in @{term R} *}

lemma (in ring) i_intersect:

  assumes "ideal I R"

  assumes "ideal J R"

  shows "ideal (I \<inter> J) R"

proof -

  interpret ideal [I R] by fact

  interpret ideal [J R] by fact

  show ?thesis

apply (intro idealI subgroup.intro)

      apply (rule is_ring)

     apply (force simp add: a_subset)

    apply (simp add: a_inv_def[symmetric])

   apply simp

  apply (simp add: a_inv_def[symmetric])

 apply (clarsimp, rule)

  apply (fast intro: ideal.I_l_closed ideal.intro prems)+

apply (clarsimp, rule)

 apply (fast intro: ideal.I_r_closed ideal.intro prems)+

done

qed

subsubsection {* Intersection of a Set of Ideals *}

text {* The intersection of any Number of Ideals is again

        an Ideal in @{term R} *}

lemma (in ring) i_Intersect:

  assumes Sideals: "\<And>I. I \<in> S \<Longrightarrow> ideal I R"

    and notempty: "S \<noteq> {}"

  shows "ideal (Inter S) R"

apply (unfold_locales)

apply (simp_all add: Inter_def INTER_def)

      apply (rule, simp) defer 1

      apply rule defer 1

      apply rule defer 1

      apply (fold a_inv_def, rule) defer 1

      apply rule defer 1

      apply rule defer 1

proof -

  fix x

  assume "\<forall>I\<in>S. x \<in> I"

  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp

  from notempty have "\<exists>I0. I0 \<in> S" by blast

  from this obtain I0 where I0S: "I0 \<in> S" by auto

  interpret ideal ["I0" "R"] by (rule Sideals[OF I0S])

  from xI[OF I0S] have "x \<in> I0" .

  from this and a_subset show "x \<in> carrier R" by fast

next

  fix x y

  assume "\<forall>I\<in>S. x \<in> I"

  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp

  assume "\<forall>I\<in>S. y \<in> I"

  hence yI: "\<And>I. I \<in> S \<Longrightarrow> y \<in> I" by simp

  fix J

  assume JS: "J \<in> S"

  interpret ideal ["J" "R"] by (rule Sideals[OF JS])

  from xI[OF JS] and yI[OF JS]

      show "x \<oplus> y \<in> J" by (rule a_closed)

next

  fix J

  assume JS: "J \<in> S"

  interpret ideal ["J" "R"] by (rule Sideals[OF JS])

  show "\<zero> \<in> J" by simp

next

  fix x

  assume "\<forall>I\<in>S. x \<in> I"

  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp

  fix J

  assume JS: "J \<in> S"

  interpret ideal ["J" "R"] by (rule Sideals[OF JS])

  from xI[OF JS]

      show "\<ominus> x \<in> J" by (rule a_inv_closed)

next

  fix x y

  assume "\<forall>I\<in>S. x \<in> I"

  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp

  assume ycarr: "y \<in> carrier R"

  fix J

  assume JS: "J \<in> S"

  interpret ideal ["J" "R"] by (rule Sideals[OF JS])

  from xI[OF JS] and ycarr

      show "y \<otimes> x \<in> J" by (rule I_l_closed)

next

  fix x y

  assume "\<forall>I\<in>S. x \<in> I"

  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp

  assume ycarr: "y \<in> carrier R"

  fix J

  assume JS: "J \<in> S"

  interpret ideal ["J" "R"] by (rule Sideals[OF JS])

  from xI[OF JS] and ycarr

      show "x \<otimes> y \<in> J" by (rule I_r_closed)

qed

subsection {* Addition of Ideals *}

lemma (in ring) add_ideals:

  assumes idealI: "ideal I R"

      and idealJ: "ideal J R"

  shows "ideal (I <+> J) R"

apply (rule ideal.intro)

  apply (rule add_additive_subgroups)

   apply (intro ideal.axioms[OF idealI])

  apply (intro ideal.axioms[OF idealJ])

 apply (rule is_ring)

apply (rule ideal_axioms.intro)

 apply (simp add: set_add_defs, clarsimp) defer 1

 apply (simp add: set_add_defs, clarsimp) defer 1

proof -

  fix x i j

  assume xcarr: "x \<in> carrier R"

     and iI: "i \<in> I"

     and jJ: "j \<in> J"

  from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]

      have c: "(i \<oplus> j) \<otimes> x = (i \<otimes> x) \<oplus> (j \<otimes> x)" by algebra

  from xcarr and iI

      have a: "i \<otimes> x \<in> I" by (simp add: ideal.I_r_closed[OF idealI])

  from xcarr and jJ

      have b: "j \<otimes> x \<in> J" by (simp add: ideal.I_r_closed[OF idealJ])

  from a b c

      show "\<exists>ha\<in>I. \<exists>ka\<in>J. (i \<oplus> j) \<otimes> x = ha \<oplus> ka" by fast

next

  fix x i j

  assume xcarr: "x \<in> carrier R"

     and iI: "i \<in> I"

     and jJ: "j \<in> J"

  from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]

      have c: "x \<otimes> (i \<oplus> j) = (x \<otimes> i) \<oplus> (x \<otimes> j)" by algebra

  from xcarr and iI

      have a: "x \<otimes> i \<in> I" by (simp add: ideal.I_l_closed[OF idealI])

  from xcarr and jJ

      have b: "x \<otimes> j \<in> J" by (simp add: ideal.I_l_closed[OF idealJ])

  from a b c

      show "\<exists>ha\<in>I. \<exists>ka\<in>J. x \<otimes> (i \<oplus> j) = ha \<oplus> ka" by fast

qed

subsection {* Ideals generated by a subset of @{term [locale=ring]

  "carrier R"} *}

subsubsection {* Generation of Ideals in General Rings *}

text {* @{term genideal} generates an ideal *}

lemma (in ring) genideal_ideal:

  assumes Scarr: "S \<subseteq> carrier R"

  shows "ideal (Idl S) R"

unfolding genideal_def

proof (rule i_Intersect, fast, simp)

  from oneideal and Scarr

  show "\<exists>I. ideal I R \<and> S \<le> I" by fast

qed

lemma (in ring) genideal_self:

  assumes "S \<subseteq> carrier R"

  shows "S \<subseteq> Idl S"

unfolding genideal_def

by fast

lemma (in ring) genideal_self':

  assumes carr: "i \<in> carrier R"

  shows "i \<in> Idl {i}"

proof -

  from carr

      have "{i} \<subseteq> Idl {i}" by (fast intro!: genideal_self)

  thus "i \<in> Idl {i}" by fast

qed

text {* @{term genideal} generates the minimal ideal *}

lemma (in ring) genideal_minimal:

  assumes a: "ideal I R"

      and b: "S \<subseteq> I"

  shows "Idl S \<subseteq> I"

unfolding genideal_def

by (rule, elim InterD, simp add: a b)

text {* Generated ideals and subsets *}

lemma (in ring) Idl_subset_ideal:

  assumes Iideal: "ideal I R"

      and Hcarr: "H \<subseteq> carrier R"

  shows "(Idl H \<subseteq> I) = (H \<subseteq> I)"

proof

  assume a: "Idl H \<subseteq> I"

  from Hcarr have "H \<subseteq> Idl H" by (rule genideal_self)

  from this and a

      show "H \<subseteq> I" by simp

next

  fix x

  assume HI: "H \<subseteq> I"

  from Iideal and HI

      have "I \<in> {I. ideal I R \<and> H \<subseteq> I}" by fast

  from this

      show "Idl H \<subseteq> I"

      unfolding genideal_def

      by fast

qed

lemma (in ring) subset_Idl_subset:

  assumes Icarr: "I \<subseteq> carrier R"

      and HI: "H \<subseteq> I"

  shows "Idl H \<subseteq> Idl I"

proof -

  from HI and genideal_self[OF Icarr] 

      have HIdlI: "H \<subseteq> Idl I" by fast

  from Icarr

      have Iideal: "ideal (Idl I) R" by (rule genideal_ideal)

  from HI and Icarr

      have "H \<subseteq> carrier R" by fast

  from Iideal and this

      have "(H \<subseteq> Idl I) = (Idl H \<subseteq> Idl I)"

      by (rule Idl_subset_ideal[symmetric])

  from HIdlI and this

      show "Idl H \<subseteq> Idl I" by simp

qed

lemma (in ring) Idl_subset_ideal':

  assumes acarr: "a \<in> carrier R" and bcarr: "b \<in> carrier R"

  shows "(Idl {a} \<subseteq> Idl {b}) = (a \<in> Idl {b})"

apply (subst Idl_subset_ideal[OF genideal_ideal[of "{b}"], of "{a}"])

  apply (fast intro: bcarr, fast intro: acarr)

apply fast

done

lemma (in ring) genideal_zero:

  "Idl {\<zero>} = {\<zero>}"

apply rule

 apply (rule genideal_minimal[OF zeroideal], simp)

apply (simp add: genideal_self')

done

lemma (in ring) genideal_one:

  "Idl {\<one>} = carrier R"

proof -

  interpret ideal ["Idl {\<one>}" "R"] by (rule genideal_ideal, fast intro: one_closed)

  show "Idl {\<one>} = carrier R"

  apply (rule, rule a_subset)

  apply (simp add: one_imp_carrier genideal_self')

  done

qed

subsubsection {* Generation of Principal Ideals in Commutative Rings *}

constdefs (structure R)

  cgenideal :: "('a, 'b) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a set"  ("PIdl\<index> _" [80] 79)

  "cgenideal R a \<equiv> { x \<otimes> a | x. x \<in> carrier R }"

text {* genhideal (?) really generates an ideal *}

lemma (in cring) cgenideal_ideal:

  assumes acarr: "a \<in> carrier R"

  shows "ideal (PIdl a) R"

apply (unfold cgenideal_def)

apply (rule idealI[OF is_ring])

   apply (rule subgroup.intro)

      apply (simp_all add: monoid_record_simps)

      apply (blast intro: acarr m_closed)

      apply clarsimp defer 1

      defer 1

      apply (fold a_inv_def, clarsimp) defer 1

      apply clarsimp defer 1

      apply clarsimp defer 1

proof -

  fix x y

  assume xcarr: "x \<in> carrier R"

     and ycarr: "y \<in> carrier R"

  note carr = acarr xcarr ycarr

  from carr

      have "x \<otimes> a \<oplus> y \<otimes> a = (x \<oplus> y) \<otimes> a" by (simp add: l_distr)

  from this and carr

      show "\<exists>z. x \<otimes> a \<oplus> y \<otimes> a = z \<otimes> a \<and> z \<in> carrier R" by fast

next

  from l_null[OF acarr, symmetric] and zero_closed

      show "\<exists>x. \<zero> = x \<otimes> a \<and> x \<in> carrier R" by fast

next

  fix x

  assume xcarr: "x \<in> carrier R"

  note carr = acarr xcarr

  from carr

      have "\<ominus> (x \<otimes> a) = (\<ominus> x) \<otimes> a" by (simp add: l_minus)

  from this and carr

      show "\<exists>z. \<ominus> (x \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R" by fast

next

  fix x y

  assume xcarr: "x \<in> carrier R"

     and ycarr: "y \<in> carrier R"

  note carr = acarr xcarr ycarr

  from carr

      have "y \<otimes> a \<otimes> x = (y \<otimes> x) \<otimes> a" by (simp add: m_assoc, simp add: m_comm)

  from this and carr

      show "\<exists>z. y \<otimes> a \<otimes> x = z \<otimes> a \<and> z \<in> carrier R" by fast

next

  fix x y

  assume xcarr: "x \<in> carrier R"

     and ycarr: "y \<in> carrier R"

  note carr = acarr xcarr ycarr

  from carr

      have "x \<otimes> (y \<otimes> a) = (x \<otimes> y) \<otimes> a" by (simp add: m_assoc)

  from this and carr

      show "\<exists>z. x \<otimes> (y \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R" by fast

qed

lemma (in ring) cgenideal_self:

  assumes icarr: "i \<in> carrier R"

  shows "i \<in> PIdl i"

unfolding cgenideal_def

proof simp

  from icarr

      have "i = \<one> \<otimes> i" by simp

  from this and icarr

      show "\<exists>x. i = x \<otimes> i \<and> x \<in> carrier R" by fast

qed

text {* @{const "cgenideal"} is minimal *}

lemma (in ring) cgenideal_minimal:

  assumes "ideal J R"

  assumes aJ: "a \<in> J"

  shows "PIdl a \<subseteq> J"

proof -

  interpret ideal [J R] by fact

  show ?thesis unfolding cgenideal_def

    apply rule

    apply clarify

    using aJ

    apply (erule I_l_closed)

    done

qed

lemma (in cring) cgenideal_eq_genideal:

  assumes icarr: "i \<in> carrier R"

  shows "PIdl i = Idl {i}"

apply rule

 apply (intro cgenideal_minimal)

  apply (rule genideal_ideal, fast intro: icarr)

 apply (rule genideal_self', fast intro: icarr)

apply (intro genideal_minimal)

 apply (rule cgenideal_ideal [OF icarr])

apply (simp, rule cgenideal_self [OF icarr])

done

lemma (in cring) cgenideal_eq_rcos:

 "PIdl i = carrier R #> i"

unfolding cgenideal_def r_coset_def

by fast

lemma (in cring) cgenideal_is_principalideal:

  assumes icarr: "i \<in> carrier R"

  shows "principalideal (PIdl i) R"

apply (rule principalidealI)

apply (rule cgenideal_ideal [OF icarr])

proof -

  from icarr

      have "PIdl i = Idl {i}" by (rule cgenideal_eq_genideal)

  from icarr and this

      show "\<exists>i'\<in>carrier R. PIdl i = Idl {i'}" by fast

qed

subsection {* Union of Ideals *}

lemma (in ring) union_genideal:

  assumes idealI: "ideal I R"

      and idealJ: "ideal J R"

  shows "Idl (I \<union> J) = I <+> J"

apply rule

 apply (rule ring.genideal_minimal)

   apply (rule R.is_ring)

  apply (rule add_ideals[OF idealI idealJ])

 apply (rule)

 apply (simp add: set_add_defs) apply (elim disjE) defer 1 defer 1

 apply (rule) apply (simp add: set_add_defs genideal_def) apply clarsimp defer 1

proof -

  fix x

  assume xI: "x \<in> I"

  have ZJ: "\<zero> \<in> J"

      by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealJ])

  from ideal.Icarr[OF idealI xI]

      have "x = x \<oplus> \<zero>" by algebra

  from xI and ZJ and this

      show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k" by fast

next

  fix x

  assume xJ: "x \<in> J"

  have ZI: "\<zero> \<in> I"

      by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealI])

  from ideal.Icarr[OF idealJ xJ]

      have "x = \<zero> \<oplus> x" by algebra

  from ZI and xJ and this

      show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k" by fast

next

  fix i j K

  assume iI: "i \<in> I"

     and jJ: "j \<in> J"

     and idealK: "ideal K R"

     and IK: "I \<subseteq> K"

     and JK: "J \<subseteq> K"

  from iI and IK

     have iK: "i \<in> K" by fast

  from jJ and JK

     have jK: "j \<in> K" by fast

  from iK and jK

     show "i \<oplus> j \<in> K" by (intro additive_subgroup.a_closed) (rule ideal.axioms[OF idealK])

qed

subsection {* Properties of Principal Ideals *}

text {* @{text "\<zero>"} generates the zero ideal *}

lemma (in ring) zero_genideal:

  shows "Idl {\<zero>} = {\<zero>}"

apply rule

apply (simp add: genideal_minimal zeroideal)

apply (fast intro!: genideal_self)

done

text {* @{text "\<one>"} generates the unit ideal *}

lemma (in ring) one_genideal:

  shows "Idl {\<one>} = carrier R"

proof -

  have "\<one> \<in> Idl {\<one>}" by (simp add: genideal_self')

  thus "Idl {\<one>} = carrier R" by (intro ideal.one_imp_carrier, fast intro: genideal_ideal)

qed

text {* The zero ideal is a principal ideal *}

corollary (in ring) zeropideal:

  shows "principalideal {\<zero>} R"

apply (rule principalidealI)

 apply (rule zeroideal)

apply (blast intro!: zero_closed zero_genideal[symmetric])

done

text {* The unit ideal is a principal ideal *}

corollary (in ring) onepideal:

  shows "principalideal (carrier R) R"

apply (rule principalidealI)

 apply (rule oneideal)

apply (blast intro!: one_closed one_genideal[symmetric])

done

text {* Every principal ideal is a right coset of the carrier *}

lemma (in principalideal) rcos_generate:

  assumes "cring R"

  shows "\<exists>x\<in>I. I = carrier R #> x"

proof -

  interpret cring [R] by fact

  from generate

      obtain i

        where icarr: "i \<in> carrier R"

        and I1: "I = Idl {i}"

      by fast+

  from icarr and genideal_self[of "{i}"]

      have "i \<in> Idl {i}" by fast

  hence iI: "i \<in> I" by (simp add: I1)

  from I1 icarr

      have I2: "I = PIdl i" by (simp add: cgenideal_eq_genideal)

  have "PIdl i = carrier R #> i"

      unfolding cgenideal_def r_coset_def

      by fast

  from I2 and this

      have "I = carrier R #> i" by simp

  from iI and this

      show "\<exists>x\<in>I. I = carrier R #> x" by fast

qed

subsection {* Prime Ideals *}

lemma (in ideal) primeidealCD:

  assumes "cring R"

  assumes notprime: "\<not> primeideal I R"

  shows "carrier R = I \<or> (\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I)"

proof (rule ccontr, clarsimp)

  interpret cring [R] by fact

  assume InR: "carrier R \<noteq> I"

     and "\<forall>a. a \<in> carrier R \<longrightarrow> (\<forall>b. a \<otimes> b \<in> I \<longrightarrow> b \<in> carrier R \<longrightarrow> a \<in> I \<or> b \<in> I)"

  hence I_prime: "\<And> a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I" by simp

  have "primeideal I R"

      apply (rule primeideal.intro [OF is_ideal is_cring])

      apply (rule primeideal_axioms.intro)

       apply (rule InR)

      apply (erule (2) I_prime)

      done

  from this and notprime

      show "False" by simp

qed

lemma (in ideal) primeidealCE:

  assumes "cring R"

  assumes notprime: "\<not> primeideal I R"

  obtains "carrier R = I"

    | "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I"

proof -

  interpret R: cring [R] by fact

  assume "carrier R = I ==> thesis"

    and "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I \<Longrightarrow> thesis"

  then show thesis using primeidealCD [OF R.is_cring notprime] by blast

qed

text {* If @{text "{\<zero>}"} is a prime ideal of a commutative ring, the ring is a domain *}

lemma (in cring) zeroprimeideal_domainI:

  assumes pi: "primeideal {\<zero>} R"

  shows "domain R"

apply (rule domain.intro, rule is_cring)

apply (rule domain_axioms.intro)

proof (rule ccontr, simp)

  interpret primeideal ["{\<zero>}" "R"] by (rule pi)

  assume "\<one> = \<zero>"

  hence "carrier R = {\<zero>}" by (rule one_zeroD)

  from this[symmetric] and I_notcarr

      show "False" by simp

next

  interpret primeideal ["{\<zero>}" "R"] by (rule pi)

  fix a b

  assume ab: "a \<otimes> b = \<zero>"

     and carr: "a \<in> carrier R" "b \<in> carrier R"

  from ab

      have abI: "a \<otimes> b \<in> {\<zero>}" by fast

  from carr and this

      have "a \<in> {\<zero>} \<or> b \<in> {\<zero>}" by (rule I_prime)

  thus "a = \<zero> \<or> b = \<zero>" by simp

qed

corollary (in cring) domain_eq_zeroprimeideal:

  shows "domain R = primeideal {\<zero>} R"

apply rule

 apply (erule domain.zeroprimeideal)

apply (erule zeroprimeideal_domainI)

done

subsection {* Maximal Ideals *}

lemma (in ideal) helper_I_closed:

  assumes carr: "a \<in> carrier R" "x \<in> carrier R" "y \<in> carrier R"

      and axI: "a \<otimes> x \<in> I"

  shows "a \<otimes> (x \<otimes> y) \<in> I"

proof -

  from axI and carr

     have "(a \<otimes> x) \<otimes> y \<in> I" by (simp add: I_r_closed)

  also from carr

     have "(a \<otimes> x) \<otimes> y = a \<otimes> (x \<otimes> y)" by (simp add: m_assoc)

  finally

     show "a \<otimes> (x \<otimes> y) \<in> I" .

qed

lemma (in ideal) helper_max_prime:

  assumes "cring R"

  assumes acarr: "a \<in> carrier R"

  shows "ideal {x\<in>carrier R. a \<otimes> x \<in> I} R"

proof -

  interpret cring [R] by fact

  show ?thesis apply (rule idealI)

    apply (rule cring.axioms[OF is_cring])

    apply (rule subgroup.intro)

    apply (simp, fast)

    apply clarsimp apply (simp add: r_distr acarr)

    apply (simp add: acarr)

    apply (simp add: a_inv_def[symmetric], clarify) defer 1

    apply clarsimp defer 1

    apply (fast intro!: helper_I_closed acarr)

  proof -

    fix x

    assume xcarr: "x \<in> carrier R"

      and ax: "a \<otimes> x \<in> I"

    from ax and acarr xcarr

    have "\<ominus>(a \<otimes> x) \<in> I" by simp

    also from acarr xcarr

    have "\<ominus>(a \<otimes> x) = a \<otimes> (\<ominus>x)" by algebra

    finally

    show "a \<otimes> (\<ominus>x) \<in> I" .

    from acarr

    have "a \<otimes> \<zero> = \<zero>" by simp

  next

    fix x y

    assume xcarr: "x \<in> carrier R"

      and ycarr: "y \<in> carrier R"

      and ayI: "a \<otimes> y \<in> I"

    from ayI and acarr xcarr ycarr

    have "a \<otimes> (y \<otimes> x) \<in> I" by (simp add: helper_I_closed)

    moreover from xcarr ycarr

    have "y \<otimes> x = x \<otimes> y" by (simp add: m_comm)

    ultimately

    show "a \<otimes> (x \<otimes> y) \<in> I" by simp

  qed

qed

text {* In a cring every maximal ideal is prime *}

lemma (in cring) maximalideal_is_prime:

  assumes "maximalideal I R"

  shows "primeideal I R"

proof -

  interpret maximalideal [I R] by fact

  show ?thesis apply (rule ccontr)

    apply (rule primeidealCE)

    apply (rule is_cring)

    apply assumption

    apply (simp add: I_notcarr)

  proof -

    assume "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I"

    from this

    obtain a b

      where acarr: "a \<in> carrier R"

      and bcarr: "b \<in> carrier R"

      and abI: "a \<otimes> b \<in> I"

      and anI: "a \<notin> I"

      and bnI: "b \<notin> I"

      by fast

    def J \<equiv> "{x\<in>carrier R. a \<otimes> x \<in> I}"

    from R.is_cring and acarr

    have idealJ: "ideal J R" unfolding J_def by (rule helper_max_prime)

    have IsubJ: "I \<subseteq> J"

    proof

      fix x

      assume xI: "x \<in> I"

      from this and acarr

      have "a \<otimes> x \<in> I" by (intro I_l_closed)

      from xI[THEN a_Hcarr] this

      show "x \<in> J" unfolding J_def by fast

    qed

    from abI and acarr bcarr

    have "b \<in> J" unfolding J_def by fast

    from bnI and this

    have JnI: "J \<noteq> I" by fast

    from acarr

    have "a = a \<otimes> \<one>" by algebra

    from this and anI

    have "a \<otimes> \<one> \<notin> I" by simp

    from one_closed and this

    have "\<one> \<notin> J" unfolding J_def by fast

    hence Jncarr: "J \<noteq> carrier R" by fast

    interpret ideal ["J" "R"] by (rule idealJ)

    have "J = I \<or> J = carrier R"

      apply (intro I_maximal)

      apply (rule idealJ)

      apply (rule IsubJ)

      apply (rule a_subset)

      done

    from this and JnI and Jncarr

    show "False" by simp

  qed

qed

subsection {* Derived Theorems Involving Ideals *}

--"A non-zero cring that has only the two trivial ideals is a field"

lemma (in cring) trivialideals_fieldI:

  assumes carrnzero: "carrier R \<noteq> {\<zero>}"

      and haveideals: "{I. ideal I R} = {{\<zero>}, carrier R}"

  shows "field R"

apply (rule cring_fieldI)

apply (rule, rule, rule)

 apply (erule Units_closed)

defer 1

  apply rule

defer 1

proof (rule ccontr, simp)

  assume zUnit: "\<zero> \<in> Units R"

  hence a: "\<zero> \<otimes> inv \<zero> = \<one>" by (rule Units_r_inv)

  from zUnit

      have "\<zero> \<otimes> inv \<zero> = \<zero>" by (intro l_null, rule Units_inv_closed)

  from a[symmetric] and this

      have "\<one> = \<zero>" by simp

  hence "carrier R = {\<zero>}" by (rule one_zeroD)

  from this and carrnzero

      show "False" by simp

next

  fix x

  assume xcarr': "x \<in> carrier R - {\<zero>}"

  hence xcarr: "x \<in> carrier R" by fast

  from xcarr'

      have xnZ: "x \<noteq> \<zero>" by fast

  from xcarr

      have xIdl: "ideal (PIdl x) R" by (intro cgenideal_ideal, fast)

  from xcarr

      have "x \<in> PIdl x" by (intro cgenideal_self, fast)

  from this and xnZ

      have "PIdl x \<noteq> {\<zero>}" by fast

  from haveideals and this

      have "PIdl x = carrier R"

      by (blast intro!: xIdl)

  hence "\<one> \<in> PIdl x" by simp

  hence "\<exists>y. \<one> = y \<otimes> x \<and> y \<in> carrier R" unfolding cgenideal_def by blast

  from this

      obtain y

        where ycarr: " y \<in> carrier R"

        and ylinv: "\<one> = y \<otimes> x"

      by fast+

  from ylinv and xcarr ycarr

      have yrinv: "\<one> = x \<otimes> y" by (simp add: m_comm)

  from ycarr and ylinv[symmetric] and yrinv[symmetric]

      have "\<exists>y \<in> carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast

  from this and xcarr

      show "x \<in> Units R"

      unfolding Units_def

      by fast

qed

lemma (in field) all_ideals:

  shows "{I. ideal I R} = {{\<zero>}, carrier R}"

apply (rule, rule)

proof -

  fix I

  assume a: "I \<in> {I. ideal I R}"

  with this

      interpret ideal ["I" "R"] by simp

  show "I \<in> {{\<zero>}, carrier R}"

  proof (cases "\<exists>a. a \<in> I - {\<zero>}")

    assume "\<exists>a. a \<in> I - {\<zero>}"

    from this

        obtain a

          where aI: "a \<in> I"

          and anZ: "a \<noteq> \<zero>"

        by fast+

    from aI[THEN a_Hcarr] anZ

        have aUnit: "a \<in> Units R" by (simp add: field_Units)

    hence a: "a \<otimes> inv a = \<one>" by (rule Units_r_inv)

    from aI and aUnit

        have "a \<otimes> inv a \<in> I" by (simp add: I_r_closed del: Units_r_inv)

    hence oneI: "\<one> \<in> I" by (simp add: a[symmetric])

    have "carrier R \<subseteq> I"

    proof

      fix x

      assume xcarr: "x \<in> carrier R"

      from oneI and this

          have "\<one> \<otimes> x \<in> I" by (rule I_r_closed)

      from this and xcarr

          show "x \<in> I" by simp

    qed

    from this and a_subset

        have "I = carrier R" by fast

    thus "I \<in> {{\<zero>}, carrier R}" by fast

  next

    assume "\<not> (\<exists>a. a \<in> I - {\<zero>})"

    hence IZ: "\<And>a. a \<in> I \<Longrightarrow> a = \<zero>" by simp

    have a: "I \<subseteq> {\<zero>}"

    proof

      fix x

      assume "x \<in> I"

      hence "x = \<zero>" by (rule IZ)

      thus "x \<in> {\<zero>}" by fast

    qed

    have "\<zero> \<in> I" by simp

    hence "{\<zero>} \<subseteq> I" by fast

    from this and a

        have "I = {\<zero>}" by fast

    thus "I \<in> {{\<zero>}, carrier R}" by fast

  qed

qed (simp add: zeroideal oneideal)

--"Jacobson Theorem 2.2"

lemma (in cring) trivialideals_eq_field:

  assumes carrnzero: "carrier R \<noteq> {\<zero>}"

  shows "({I. ideal I R} = {{\<zero>}, carrier R}) = field R"

by (fast intro!: trivialideals_fieldI[OF carrnzero] field.all_ideals)