(*

   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, assumption)

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

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

 done

 lemma (in ideal) is_ideal:

   "ideal I R"

 by -

 lemma idealI:

   includes ring

   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"

 by (intro ideal.intro ideal_axioms.intro additive_subgroupI, assumption+)

 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 -

 lemma principalidealI:

   includes ideal

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

   shows "principalideal I R"

 by (intro principalideal.intro principalideal_axioms.intro, assumption+)

 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 -

 lemma maximalidealI:

   includes ideal

   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"

 by (intro maximalideal.intro maximalideal_axioms.intro, assumption+)

 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 -

 lemma primeidealI:

   includes ideal

   includes cring

   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"

 by (intro primeideal.intro primeideal_axioms.intro, assumption+)

 lemma primeidealI2:

   includes additive_subgroup I R

   includes cring

   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"

 apply (intro_locales)

  apply (intro ideal_axioms.intro, assumption+)

 apply (intro primeideal_axioms.intro, assumption+)

 done

 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,assumption)

  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"

 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:

   includes ideal I R

   includes ideal J R

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

 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

 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 assumption

 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"

   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:

   includes ideal J R

   assumes aJ: "a \<in> J"

   shows "PIdl a \<subseteq> J"

 unfolding cgenideal_def

 by (rule, clarify, rule I_l_closed)

 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, assumption)

 apply (simp, rule cgenideal_self, assumption)

 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, assumption)

 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 assumption

   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:

   includes cring

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

 proof -

   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:

   includes cring

   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)

   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, assumption+)

       by (rule primeideal_axioms.intro, rule InR, erule I_prime)

   from this and notprime

       show "False" by simp

 qed

 lemma (in ideal) primeidealCE:

   includes cring

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

     and elim1: "carrier R = I \<Longrightarrow> P"

     and elim2: "(\<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> P"

   shows "P"

 proof -

   from notprime

   have "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)"

       by (intro primeidealCD)

   thus "P"

       apply (rule disjE)

       by (erule elim1, erule elim2)

 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, assumption+)

 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:

   includes cring

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

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

 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

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

 lemma (in cring) maximalideal_is_prime:

   includes maximalideal

   shows "primeideal I R"

 apply (rule ccontr)

 apply (rule primeidealCE)

   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 acarr

       have idealJ: "ideal J R" by (unfold J_def, intro 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" by (unfold J_def, fast)

   qed

   from abI and acarr bcarr

       have "b \<in> J" by (unfold J_def, 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" by (unfold J_def, 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

 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)

     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)

 text {* Like zeroprimeideal for domains *}

 lemma (in field) zeromaximalideal:

   "maximalideal {\<zero>} R"

 apply (rule maximalidealI)

   apply (rule zeroideal)

 proof-

   from one_not_zero

       have "\<one> \<notin> {\<zero>}" by simp

   from this and one_closed

       show "carrier R \<noteq> {\<zero>}" by fast

 next

   fix J

   assume Jideal: "ideal J R"

   hence "J \<in> {I. ideal I R}"

       by fast

   from this and all_ideals

       show "J = {\<zero>} \<or> J = carrier R" by simp

 qed

 lemma (in cring) zeromaximalideal_fieldI:

   assumes zeromax: "maximalideal {\<zero>} R"

   shows "field R"

 apply (rule trivialideals_fieldI, rule maximalideal.I_notcarr[OF zeromax])

 apply rule apply clarsimp defer 1

  apply (simp add: zeroideal oneideal)

 proof -

   fix J

   assume Jn0: "J \<noteq> {\<zero>}"

      and idealJ: "ideal J R"

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

   have "{\<zero>} \<subseteq> J" by (rule ccontr, simp)

   from zeromax and idealJ and this and a_subset

       have "J = {\<zero>} \<or> J = carrier R" by (rule maximalideal.I_maximal)

   from this and Jn0

       show "J = carrier R" by simp

 qed

 lemma (in cring) zeromaximalideal_eq_field:

   "maximalideal {\<zero>} R = field R"

 apply rule

  apply (erule zeromaximalideal_fieldI)

 apply (erule field.zeromaximalideal)

 done

 end