Theory Ideal

(*  Title:      HOL/Algebra/Ideal.thy
    Author:     Stephan Hohe, TU Muenchen
*)

theory Ideal
imports Ring AbelCoset
begin

section ‹Ideals›

subsection ‹Definitions›

subsubsection ‹General definition›

locale ideal = additive_subgroup I R + ring R for I and R (structure) +
  assumes I_l_closed: "⟦a ∈ I; x ∈ carrier R⟧ ⟹ x ⊗ a ∈ I"
      and I_r_closed: "⟦a ∈ I; x ∈ carrier R⟧ ⟹ a ⊗ x ∈ I"

sublocale ideal ⊆ abelian_subgroup I R
proof (intro abelian_subgroupI3 abelian_group.intro)
  show "additive_subgroup I R"
    by (simp add: is_additive_subgroup)
  show "abelian_monoid R"
    by (simp add: abelian_monoid_axioms)
  show "abelian_group_axioms R"
    using abelian_group_def is_abelian_group by blast
qed

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 (add_monoid R)"
    and I_l_closed: "⋀a x. ⟦a ∈ I; x ∈ carrier R⟧ ⟹ x ⊗ a ∈ I"
    and I_r_closed: "⋀a x. ⟦a ∈ I; x ∈ carrier R⟧ ⟹ a ⊗ x ∈ I"
  shows "ideal I R"
proof -
  interpret ring R by fact
  show ?thesis  
    by (auto simp: ideal.intro ideal_axioms.intro additive_subgroupI a_subgroup ring_axioms I_l_closed I_r_closed)
qed


subsubsection (in ring) ‹Ideals Generated by a Subset of \<^term>‹carrier R››

definition genideal :: "_ ⇒ 'a set ⇒ 'a set"
    (‹(‹open_block notation=‹prefix Idl››Idlı _)› [80] 79)
  where "Idl⇘R⇙ S = ⋂{I. ideal I R ∧ S ⊆ I}"

subsubsection ‹Principal Ideals›

locale principalideal = ideal +
  assumes generate: "∃i ∈ carrier R. I = Idl {i}"

lemma (in principalideal) is_principalideal: "principalideal I R"
  by (rule principalideal_axioms)

lemma principalidealI:
  fixes R (structure)
  assumes "ideal I R"
    and generate: "∃i ∈ 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

(* NEW ====== *)
lemma (in ideal) rcos_const_imp_mem:
  assumes "i ∈ carrier R" and "I +> i = I" shows "i ∈ I"
  using additive_subgroup.zero_closed[OF ideal.axioms(1)[OF ideal_axioms]] assms
  by (force simp add: a_r_coset_def')
(* ========== *)

(* NEW ====== *)
lemma (in ring) a_rcos_zero:
  assumes "ideal I R" "i ∈ I" shows "I +> i = I"
  using abelian_subgroupI3[OF ideal.axioms(1) is_abelian_group]
  by (simp add: abelian_subgroup.a_rcos_const assms)
(* ========== *)

(* NEW ====== *)
lemma (in ring) ideal_is_normal:
  assumes "ideal I R" shows "I ⊲ (add_monoid R)"
  using abelian_subgroup.a_normal[OF abelian_subgroupI3[OF ideal.axioms(1)]]
        abelian_group_axioms assms
  by auto 
(* ========== *)

(* NEW ====== *)
lemma (in ideal) a_rcos_sum:
  assumes "a ∈ carrier R" and "b ∈ carrier R" shows "(I +> a) <+> (I +> b) = I +> (a ⊕ b)"
  using normal.rcos_sum[OF ideal_is_normal[OF ideal_axioms]] assms
  unfolding set_add_def a_r_coset_def by simp
(* ========== *)

(* NEW ====== *)
lemma (in ring) set_add_comm:
  assumes "I ⊆ carrier R" "J ⊆ carrier R" shows "I <+> J = J <+> I"
proof -
  have "I <+> J ⊆ J <+> I" if "I ⊆ carrier R" "J ⊆ carrier R" for I J
    using that a_comm unfolding set_add_def' by (auto, blast)
  thus ?thesis
    using assms by auto
qed
(* ========== *)


subsubsection ‹Maximal Ideals›

locale maximalideal = ideal +
  assumes I_notcarr: "carrier R ≠ I"
    and I_maximal: "⟦ideal J R; I ⊆ J; J ⊆ carrier R⟧ ⟹ (J = I) ∨ (J = carrier R)"

lemma (in maximalideal) is_maximalideal: "maximalideal I R"
  by (rule maximalideal_axioms)

lemma maximalidealI:
  fixes R
  assumes "ideal I R"
    and I_notcarr: "carrier R ≠ I"
    and I_maximal: "⋀J. ⟦ideal J R; I ⊆ J; J ⊆ carrier R⟧ ⟹ (J = I) ∨ (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


subsubsection ‹Prime Ideals›

locale primeideal = ideal + cring +
  assumes I_notcarr: "carrier R ≠ I"
    and I_prime: "⟦a ∈ carrier R; b ∈ carrier R; a ⊗ b ∈ I⟧ ⟹ a ∈ I ∨ b ∈ I"

lemma (in primeideal) primeideal: "primeideal I R"
  by (rule primeideal_axioms)

lemma primeidealI:
  fixes R (structure)
  assumes "ideal I R"
    and "cring R"
    and I_notcarr: "carrier R ≠ I"
    and I_prime: "⋀a b. ⟦a ∈ carrier R; b ∈ carrier R; a ⊗ b ∈ I⟧ ⟹ a ∈ I ∨ b ∈ 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"
    and "cring R"
    and I_l_closed: "⋀a x. ⟦a ∈ I; x ∈ carrier R⟧ ⟹ x ⊗ a ∈ I"
    and I_r_closed: "⋀a x. ⟦a ∈ I; x ∈ carrier R⟧ ⟹ a ⊗ x ∈ I"
    and I_notcarr: "carrier R ≠ I"
    and I_prime: "⋀a b. ⟦a ∈ carrier R; b ∈ carrier R; a ⊗ b ∈ I⟧ ⟹ a ∈ I ∨ b ∈ 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)
    by (simp add: I_notcarr I_prime primeideal_axioms.intro)
qed


subsection ‹Special Ideals›

lemma (in ring) zeroideal: "ideal {𝟬} R"
  by (intro idealI subgroup.intro) (simp_all add: ring_axioms)

lemma (in ring) oneideal: "ideal (carrier R) R"
  by (rule idealI) (auto intro: ring_axioms add.subgroupI)

lemma (in "domain") zeroprimeideal: "primeideal {𝟬} R"
proof -
  have "carrier R ≠ {𝟬}"
    by (simp add: carrier_one_not_zero)
  then show ?thesis
    by (metis (no_types, lifting) domain_axioms domain_def integral primeidealI singleton_iff zeroideal)
qed


subsection ‹General Ideal Properties›

lemma (in ideal) one_imp_carrier:
  assumes I_one_closed: "𝟭 ∈ I"
  shows "I = carrier R"
proof
  show "carrier R ⊆ I"
    using I_r_closed assms by fastforce
  show "I ⊆ carrier R"
    by (rule a_subset)
qed

lemma (in ideal) Icarr:
  assumes iI: "i ∈ I"
  shows "i ∈ carrier R"
  using iI by (rule a_Hcarr)

lemma (in ring) quotient_eq_iff_same_a_r_cos:
  assumes "ideal I R" and "a ∈ carrier R" and "b ∈ carrier R"
  shows "a ⊖ b ∈ I ⟷ I +> a = I +> b"
proof
  assume "I +> a = I +> b"
  then obtain i where "i ∈ I" and "𝟬 ⊕ a = i ⊕ b"
    using additive_subgroup.zero_closed[OF ideal.axioms(1)[OF assms(1)]] assms(2)
    unfolding a_r_coset_def' by blast
  hence "a ⊖ b = i"
    using assms(2-3) by (metis a_minus_def add.inv_solve_right assms(1) ideal.Icarr l_zero)
  with ‹i ∈ I› show "a ⊖ b ∈ I"
    by simp
next
  assume "a ⊖ b ∈ I"
  then obtain i where "i ∈ I" and "a = i ⊕ b"
    using ideal.Icarr[OF assms(1)] assms(2-3)
    by (metis a_minus_def add.inv_solve_right)
  hence "I +> a = (I +> i) +> b"
    using ideal.Icarr[OF assms(1)] assms(3)
    by (simp add: a_coset_add_assoc subsetI)
  with ‹i ∈ I› show "I +> a = I +> b"
    using a_rcos_zero[OF assms(1)] by simp
qed


subsection ‹Intersection of Ideals›

paragraph ‹Intersection of two ideals›
text ‹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 ∩ J) R"
proof -
  interpret ideal I R by fact
  interpret ideal J R by fact
  have IJ: "I ∩ J ⊆ carrier R"
    by (force simp: a_subset)
  show ?thesis
    apply (intro idealI subgroup.intro)
    apply (simp_all add: IJ ring_axioms I_l_closed assms ideal.I_l_closed ideal.I_r_closed flip: a_inv_def)
    done
qed

text ‹The intersection of any Number of Ideals is again an Ideal in \<^term>‹R››

lemma (in ring) i_Intersect:
  assumes Sideals: "⋀I. I ∈ S ⟹ ideal I R" and notempty: "S ≠ {}"
  shows "ideal (⋂S) R"
proof -
  have "x ⊕ y ∈ J" if "∀I∈S. x ∈ I" "∀I∈S. y ∈ I" and JS: "J ∈ S" for x y J
  proof -
    interpret ideal J R by (rule Sideals[OF JS])
    show ?thesis by (simp add: JS ‹∀I∈S. x ∈ I› ‹∀I∈S. y ∈ I›)
  qed
  moreover have "𝟬 ∈ J" if "J ∈ S" for J
    by (simp add: that Sideals additive_subgroup.zero_closed ideal.axioms(1)) 
  moreover have "⊖ x ∈ J" if "∀I∈S. x ∈ I" and JS: "J ∈ S" for x J
  proof -
    interpret ideal J R by (rule Sideals[OF JS])
    show ?thesis by (simp add: JS ‹∀I∈S. x ∈ I›)
  qed
  moreover have "y ⊗ x ∈ J" "x ⊗ y ∈ J"
    if "∀I∈S. x ∈ I" and ycarr: "y ∈ carrier R" and JS: "J ∈ S" for x y J
  proof -
    interpret ideal J R by (rule Sideals[OF JS])
    show "y ⊗ x ∈ J" "x ⊗ y ∈ J" using I_l_closed I_r_closed JS ‹∀I∈S. x ∈ I› ycarr by blast+
  qed
  moreover have "x ∈ carrier R" if "∀I∈S. x ∈ I" for x
    proof -
    obtain I0 where I0S: "I0 ∈ S"
      using notempty by blast
    interpret ideal I0 R by (rule Sideals[OF I0S])
    have "x ∈ I0"
      by (simp add: I0S ‹∀I∈S. x ∈ I›) 
    with a_subset show ?thesis by fast
  qed
  ultimately show ?thesis
    by unfold_locales (auto simp: Inter_eq simp flip: a_inv_def)
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"
proof (rule ideal.intro)
  show "additive_subgroup (I <+> J) R"
    by (intro ideal.axioms[OF idealI] ideal.axioms[OF idealJ] add_additive_subgroups)
  show "ring R"
    by (rule ring_axioms)
  show "ideal_axioms (I <+> J) R"
  proof -
    have "∃h∈I. ∃k∈J. (i ⊕ j) ⊗ x = h ⊕ k"
      if xcarr: "x ∈ carrier R" and iI: "i ∈ I" and jJ: "j ∈ J" for x i j
      using xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
      by (meson iI ideal.I_r_closed idealJ jJ l_distr local.idealI)
    moreover have "∃h∈I. ∃k∈J. x ⊗ (i ⊕ j) = h ⊕ k"
      if xcarr: "x ∈ carrier R" and iI: "i ∈ I" and jJ: "j ∈ J" for x i j
      using xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
        by (meson iI ideal.I_l_closed idealJ jJ local.idealI r_distr)
    ultimately show "ideal_axioms (I <+> J) R"
      by (intro ideal_axioms.intro) (auto simp: set_add_defs)
  qed
qed

subsection (in ring) ‹Ideals generated by a subset of \<^term>‹carrier R››

text ‹\<^term>‹genideal› generates an ideal›
lemma (in ring) genideal_ideal:
  assumes Scarr: "S ⊆ carrier R"
  shows "ideal (Idl S) R"
unfolding genideal_def
proof (rule i_Intersect, fast, simp)
  from oneideal and Scarr
  show "∃I. ideal I R ∧ S ≤ I" by fast
qed

lemma (in ring) genideal_self:
  assumes "S ⊆ carrier R"
  shows "S ⊆ Idl S"
  unfolding genideal_def by fast

lemma (in ring) genideal_self':
  assumes carr: "i ∈ carrier R"
  shows "i ∈ Idl {i}"
  by (simp add: genideal_def)

text ‹\<^term>‹genideal› generates the minimal ideal›
lemma (in ring) genideal_minimal:
  assumes "ideal I R" "S ⊆ I"
  shows "Idl S ⊆ I"
  unfolding genideal_def by rule (elim InterD, simp add: assms)

text ‹Generated ideals and subsets›
lemma (in ring) Idl_subset_ideal:
  assumes Iideal: "ideal I R"
    and Hcarr: "H ⊆ carrier R"
  shows "(Idl H ⊆ I) = (H ⊆ I)"
proof
  assume a: "Idl H ⊆ I"
  from Hcarr have "H ⊆ Idl H" by (rule genideal_self)
  with a show "H ⊆ I" by simp
next
  fix x
  assume "H ⊆ I"
  with Iideal have "I ∈ {I. ideal I R ∧ H ⊆ I}" by fast
  then show "Idl H ⊆ I" unfolding genideal_def by fast
qed

lemma (in ring) subset_Idl_subset:
  assumes Icarr: "I ⊆ carrier R"
    and HI: "H ⊆ I"
  shows "Idl H ⊆ Idl I"
proof -
  from Icarr have Iideal: "ideal (Idl I) R"
    by (rule genideal_ideal)
  from HI and Icarr have "H ⊆ carrier R"
    by fast
  with Iideal have "(H ⊆ Idl I) = (Idl H ⊆ Idl I)"
    by (rule Idl_subset_ideal[symmetric])
  then show "Idl H ⊆ Idl I"
    by (meson HI Icarr genideal_self order_trans)
qed

lemma (in ring) Idl_subset_ideal':
  assumes acarr: "a ∈ carrier R" and bcarr: "b ∈ carrier R"
  shows "Idl {a} ⊆ Idl {b} ⟷ a ∈ Idl {b}"
proof -
  have "Idl {a} ⊆ Idl {b} ⟷ {a} ⊆ Idl {b}"
    by (simp add: Idl_subset_ideal acarr bcarr genideal_ideal)
  also have "… ⟷ a ∈ Idl {b}"
    by blast
  finally show ?thesis .
qed

lemma (in ring) genideal_zero: "Idl {𝟬} = {𝟬}"
proof
  show "Idl {𝟬} ⊆ {𝟬}"
    by (simp add: genideal_minimal zeroideal)
  show "{𝟬} ⊆ Idl {𝟬}"
    by (simp add: genideal_self')
qed

lemma (in ring) genideal_one: "Idl {𝟭} = carrier R"
proof -
  interpret ideal "Idl {𝟭}" "R" by (rule genideal_ideal) fast
  show "Idl {𝟭} = carrier R"
    using genideal_self' one_imp_carrier by blast
qed


text ‹Generation of Principal Ideals in Commutative Rings›

definition cgenideal :: "_ ⇒ 'a ⇒ 'a set"
    (‹(‹open_block notation=‹prefix PIdl››PIdlı _)› [80] 79)
  where "PIdl⇘R⇙ a = {x ⊗⇘R⇙ a | x. x ∈ carrier R}"

text ‹genhideal (?) really generates an ideal›
lemma (in cring) cgenideal_ideal:
  assumes acarr: "a ∈ carrier R"
  shows "ideal (PIdl a) R"
  unfolding cgenideal_def
proof (intro subgroup.intro idealI[OF ring_axioms], simp_all)
  show "{x ⊗ a |x. x ∈ carrier R} ⊆ carrier R"
    by (blast intro: acarr)
  show "⋀x y. ⟦∃u. x = u ⊗ a ∧ u ∈ carrier R; ∃x. y = x ⊗ a ∧ x ∈ carrier R⟧
              ⟹ ∃v. x ⊕ y = v ⊗ a ∧ v ∈ carrier R"
    by (metis assms cring.cring_simprules(1) is_cring l_distr)
  show "∃x. 𝟬 = x ⊗ a ∧ x ∈ carrier R"
    by (metis assms l_null zero_closed)
  show "⋀x. ∃u. x = u ⊗ a ∧ u ∈ carrier R 
            ⟹ ∃v. inv⇘add_monoid R⇙ x = v ⊗ a ∧ v ∈ carrier R"
    by (metis a_inv_def add.inv_closed assms l_minus)
  show "⋀b x. ⟦∃x. b = x ⊗ a ∧ x ∈ carrier R; x ∈ carrier R⟧
       ⟹ ∃z. x ⊗ b = z ⊗ a ∧ z ∈ carrier R"
    by (metis assms m_assoc m_closed)
  show "⋀b x. ⟦∃x. b = x ⊗ a ∧ x ∈ carrier R; x ∈ carrier R⟧
       ⟹ ∃z. b ⊗ x = z ⊗ a ∧ z ∈ carrier R"
    by (metis assms m_assoc m_comm m_closed)
qed

lemma (in ring) cgenideal_self:
  assumes icarr: "i ∈ carrier R"
  shows "i ∈ PIdl i"
  unfolding cgenideal_def
proof simp
  from icarr have "i = 𝟭 ⊗ i"
    by simp
  with icarr show "∃x. i = x ⊗ i ∧ x ∈ carrier R"
    by fast
qed

text ‹\<^const>‹cgenideal› is minimal›

lemma (in ring) cgenideal_minimal:
  assumes "ideal J R"
  assumes aJ: "a ∈ J"
  shows "PIdl a ⊆ J"
proof -
  interpret ideal J R by fact
  show ?thesis
    unfolding cgenideal_def
    using I_l_closed aJ by blast
qed

lemma (in cring) cgenideal_eq_genideal:
  assumes icarr: "i ∈ carrier R"
  shows "PIdl i = Idl {i}"
proof
  show "PIdl i ⊆ Idl {i}"
    by (simp add: cgenideal_minimal genideal_ideal genideal_self' icarr)
  show "Idl {i} ⊆ PIdl i"
    by (simp add: cgenideal_ideal cgenideal_self genideal_minimal icarr)
qed

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 "i ∈ carrier R"
  shows "principalideal (PIdl i) R"
proof -
  have "∃i'∈carrier R. PIdl i = Idl {i'}"
    using cgenideal_eq_genideal assms by auto
  then show ?thesis
    by (simp add: cgenideal_ideal assms principalidealI)
qed


subsection ‹Union of Ideals›

lemma (in ring) union_genideal:
  assumes idealI: "ideal I R" and idealJ: "ideal J R"
  shows "Idl (I ∪ J) = I <+> J"
proof
  show "Idl (I ∪ J) ⊆ I <+> J"
  proof (rule ring.genideal_minimal [OF ring_axioms])
    show "ideal (I <+> J) R"
      by (rule add_ideals[OF idealI idealJ])
    have "⋀x. x ∈ I ⟹ ∃xa∈I. ∃xb∈J. x = xa ⊕ xb"
      by (metis additive_subgroup.zero_closed ideal.Icarr idealJ ideal_def local.idealI r_zero)
    moreover have "⋀x. x ∈ J ⟹ ∃xa∈I. ∃xb∈J. x = xa ⊕ xb"
      by (metis additive_subgroup.zero_closed ideal.Icarr idealJ ideal_def l_zero local.idealI)
    ultimately show "I ∪ J ⊆ I <+> J"
      by (auto simp: set_add_defs) 
  qed
next
  show "I <+> J ⊆ Idl (I ∪ J)"
    by (auto simp: set_add_defs genideal_def additive_subgroup.a_closed ideal_def subsetD)
qed

subsection ‹Properties of Principal Ideals›

text ‹The zero ideal is a principal ideal›
corollary (in ring) zeropideal: "principalideal {𝟬} R"
  using genideal_zero principalidealI zeroideal by blast

text ‹The unit ideal is a principal ideal›
corollary (in ring) onepideal: "principalideal (carrier R) R"
  using genideal_one oneideal principalidealI by blast

text ‹Every principal ideal is a right coset of the carrier›
lemma (in principalideal) rcos_generate:
  assumes "cring R"
  shows "∃x∈I. I = carrier R #> x"
proof -
  interpret cring R by fact
  from generate obtain i where icarr: "i ∈ carrier R" and I1: "I = Idl {i}"
    by fast+
  then have "I = PIdl i"
    by (simp add: cgenideal_eq_genideal)
  moreover have "i ∈ I"
    by (simp add: I1 genideal_self' icarr)
  moreover have "PIdl i = carrier R #> i"
    unfolding cgenideal_def r_coset_def by fast
  ultimately show "∃x∈I. I = carrier R #> x"
    by fast
qed


text ‹This next lemma would be trivial if placed in a theory that imports QuotRing,
      but it makes more sense to have it here (easier to find and coherent with the
      previous developments).›

lemma (in cring) cgenideal_prod: ✐‹contributor ‹Paulo Emílio de Vilhena››
  assumes "a ∈ carrier R" "b ∈ carrier R"
  shows "(PIdl a) <#> (PIdl b) = PIdl (a ⊗ b)"
proof -
  have "(carrier R #> a) <#> (carrier R #> b) = carrier R #> (a ⊗ b)"
  proof
    show "(carrier R #> a) <#> (carrier R #> b) ⊆ carrier R #> a ⊗ b"
    proof
      fix x assume "x ∈ (carrier R #> a) <#> (carrier R #> b)"
      then obtain r1 r2 where r1: "r1 ∈ carrier R" and r2: "r2 ∈ carrier R"
                          and "x = (r1 ⊗ a) ⊗ (r2 ⊗ b)"
        unfolding set_mult_def r_coset_def by blast
      hence "x = (r1 ⊗ r2) ⊗ (a ⊗ b)"
        by (simp add: assms local.ring_axioms m_lcomm ring.ring_simprules(11))
      thus "x ∈ carrier R #> a ⊗ b"
        unfolding r_coset_def using r1 r2 assms by blast 
    qed
  next
    show "carrier R #> a ⊗ b ⊆ (carrier R #> a) <#> (carrier R #> b)"
    proof
      fix x assume "x ∈ carrier R #> a ⊗ b"
      then obtain r where r: "r ∈ carrier R" "x = r ⊗ (a ⊗ b)"
        unfolding r_coset_def by blast
      hence "x = (r ⊗ a) ⊗ (𝟭 ⊗ b)"
        using assms by (simp add: m_assoc)
      thus "x ∈ (carrier R #> a) <#> (carrier R #> b)"
        unfolding set_mult_def r_coset_def using assms r by blast
    qed
  qed
  thus ?thesis
    using cgenideal_eq_rcos[of a] cgenideal_eq_rcos[of b] cgenideal_eq_rcos[of "a ⊗ b"] by simp
qed


subsection ‹Prime Ideals›

lemma (in ideal) primeidealCD:
  assumes "cring R"
  assumes notprime: "¬ primeideal I R"
  shows "carrier R = I ∨ (∃a b. a ∈ carrier R ∧ b ∈ carrier R ∧ a ⊗ b ∈ I ∧ a ∉ I ∧ b ∉ I)"
proof (rule ccontr, clarsimp)
  interpret cring R by fact
  assume InR: "carrier R ≠ I"
    and "∀a. a ∈ carrier R ⟶ (∀b. a ⊗ b ∈ I ⟶ b ∈ carrier R ⟶ a ∈ I ∨ b ∈ I)"
  then have I_prime: "⋀ a b. ⟦a ∈ carrier R; b ∈ carrier R; a ⊗ b ∈ I⟧ ⟹ a ∈ I ∨ b ∈ I"
    by simp
  have "primeideal I R"
    by (simp add: I_prime InR is_cring is_ideal primeidealI)
  with notprime show False by simp
qed

lemma (in ideal) primeidealCE:
  assumes "cring R"
  assumes notprime: "¬ primeideal I R"
  obtains "carrier R = I"
    | "∃a b. a ∈ carrier R ∧ b ∈ carrier R ∧ a ⊗ b ∈ I ∧ a ∉ I ∧ b ∉ I"
proof -
  interpret R: cring R by fact
  assume "carrier R = I ==> thesis"
    and "∃a b. a ∈ carrier R ∧ b ∈ carrier R ∧ a ⊗ b ∈ I ∧ a ∉ I ∧ b ∉ I ⟹ thesis"
  then show thesis using primeidealCD [OF R.is_cring notprime] by blast
qed

text ‹If ‹{𝟬}› is a prime ideal of a commutative ring, the ring is a domain›
lemma (in cring) zeroprimeideal_domainI:
  assumes pi: "primeideal {𝟬} R"
  shows "domain R"
proof (intro domain.intro is_cring domain_axioms.intro)
  show "𝟭 ≠ 𝟬"
    using genideal_one genideal_zero pi primeideal.I_notcarr by force
  show "a = 𝟬 ∨ b = 𝟬" if ab: "a ⊗ b = 𝟬" and carr: "a ∈ carrier R" "b ∈ carrier R" for a b
  proof -
    interpret primeideal "{𝟬}" "R" by (rule pi)
    show "a = 𝟬 ∨ b = 𝟬"
      using I_prime ab carr by blast
  qed
qed

corollary (in cring) domain_eq_zeroprimeideal: "domain R = primeideal {𝟬} R"
  using domain.zeroprimeideal zeroprimeideal_domainI by blast


subsection ‹Maximal Ideals›

lemma (in ideal) helper_I_closed:
  assumes carr: "a ∈ carrier R" "x ∈ carrier R" "y ∈ carrier R"
    and axI: "a ⊗ x ∈ I"
  shows "a ⊗ (x ⊗ y) ∈ I"
proof -
  from axI and carr have "(a ⊗ x) ⊗ y ∈ I"
    by (simp add: I_r_closed)
  also from carr have "(a ⊗ x) ⊗ y = a ⊗ (x ⊗ y)"
    by (simp add: m_assoc)
  finally show "a ⊗ (x ⊗ y) ∈ I" .
qed

lemma (in ideal) helper_max_prime:
  assumes "cring R"
  assumes acarr: "a ∈ carrier R"
  shows "ideal {x∈carrier R. a ⊗ x ∈ I} R"
proof -
  interpret cring R by fact
  show ?thesis 
  proof (rule idealI, simp_all)
    show "ring R"
      by (simp add: local.ring_axioms)
    show "subgroup {x ∈ carrier R. a ⊗ x ∈ I} (add_monoid R)"
      by (rule subgroup.intro) (auto simp: r_distr acarr r_minus simp flip: a_inv_def)
    show "⋀b x. ⟦b ∈ carrier R ∧ a ⊗ b ∈ I; x ∈ carrier R⟧
                 ⟹ a ⊗ (x ⊗ b) ∈ I"
      using acarr helper_I_closed m_comm by auto
    show "⋀b x. ⟦b ∈ carrier R ∧ a ⊗ b ∈ I; x ∈ carrier R⟧
                ⟹ a ⊗ (b ⊗ x) ∈ I"
      by (simp add: acarr helper_I_closed)
  qed
qed

text ‹In a cring every maximal ideal is prime›
lemma (in cring) maximalideal_prime:
  assumes "maximalideal I R"
  shows "primeideal I R"
proof -
  interpret maximalideal I R by fact
  show ?thesis 
  proof (rule ccontr)
    assume neg: "¬ primeideal I R"
    then obtain a b where acarr: "a ∈ carrier R" and bcarr: "b ∈ carrier R"
      and abI: "a ⊗ b ∈ I" and anI: "a ∉ I" and bnI: "b ∉ I" 
      using primeidealCE [OF is_cring]
      by (metis I_notcarr)
    define J where "J = {x∈carrier R. a ⊗ x ∈ I}"
    from is_cring and acarr have idealJ: "ideal J R"
      unfolding J_def by (rule helper_max_prime)
    have IsubJ: "I ⊆ J"
      using I_l_closed J_def a_Hcarr acarr by blast
    from abI and acarr bcarr have "b ∈ J"
      unfolding J_def by fast
    with bnI have JnI: "J ≠ I" by fast
    have "𝟭 ∉ J"
      unfolding J_def by (simp add: acarr anI)
    then have Jncarr: "J ≠ carrier R" by fast
    interpret ideal J R by (rule idealJ)    
    have "J = I ∨ J = carrier R"
      by (simp add: I_maximal IsubJ a_subset is_ideal)
    with JnI and Jncarr show False by simp
  qed
qed


subsection ‹Derived Theorems›

text ‹A non-zero cring that has only the two trivial ideals is a field›
lemma (in cring) trivialideals_fieldI:
  assumes carrnzero: "carrier R ≠ {𝟬}"
    and haveideals: "{I. ideal I R} = {{𝟬}, carrier R}"
  shows "field R"
proof (intro cring_fieldI equalityI)
  show "Units R ⊆ carrier R - {𝟬}"
    by (metis Diff_empty Units_closed Units_r_inv_ex carrnzero l_null one_zeroD subsetI subset_Diff_insert)
  show "carrier R - {𝟬} ⊆ Units R"
  proof
    fix x
    assume xcarr': "x ∈ carrier R - {𝟬}"
    then have xcarr: "x ∈ carrier R" and xnZ: "x ≠ 𝟬" by auto
    from xcarr have xIdl: "ideal (PIdl x) R"
      by (intro cgenideal_ideal) fast
    have "PIdl x ≠ {𝟬}"
      using xcarr xnZ cgenideal_self by blast 
    with haveideals have "PIdl x = carrier R"
      by (blast intro!: xIdl)
    then have "𝟭 ∈ PIdl x" by simp
    then have "∃y. 𝟭 = y ⊗ x ∧ y ∈ carrier R"
      unfolding cgenideal_def by blast
    then obtain y where ycarr: " y ∈ carrier R" and ylinv: "𝟭 = y ⊗ x"
      by fast    
    have "∃y ∈ carrier R. y ⊗ x = 𝟭 ∧ x ⊗ y = 𝟭"
      using m_comm xcarr ycarr ylinv by auto
    with xcarr show "x ∈ Units R"
      unfolding Units_def by fast
  qed
qed

lemma (in field) all_ideals: "{I. ideal I R} = {{𝟬}, carrier R}"
proof (intro equalityI subsetI)
  fix I
  assume a: "I ∈ {I. ideal I R}"
  then interpret ideal I R by simp

  show "I ∈ {{𝟬}, carrier R}"
  proof (cases "∃a. a ∈ I - {𝟬}")
    case True
    then obtain a where aI: "a ∈ I" and anZ: "a ≠ 𝟬"
      by fast+
    have aUnit: "a ∈ Units R"
      by (simp add: aI anZ field_Units)
    then have a: "a ⊗ inv a = 𝟭" by (rule Units_r_inv)
    from aI and aUnit have "a ⊗ inv a ∈ I"
      by (simp add: I_r_closed del: Units_r_inv)
    then have oneI: "𝟭 ∈ I" by (simp add: a[symmetric])
    have "carrier R ⊆ I"
      using oneI one_imp_carrier by auto
    with a_subset have "I = carrier R" by fast
    then show "I ∈ {{𝟬}, carrier R}" by fast
  next
    case False
    then have IZ: "⋀a. a ∈ I ⟹ a = 𝟬" by simp
    have a: "I ⊆ {𝟬}"
      using False by auto
    have "𝟬 ∈ I" by simp
    with a have "I = {𝟬}" by fast
    then show "I ∈ {{𝟬}, carrier R}" by fast
  qed
qed (auto simp: zeroideal oneideal)

―‹"Jacobson Theorem 2.2"›
lemma (in cring) trivialideals_eq_field:
  assumes carrnzero: "carrier R ≠ {𝟬}"
  shows "({I. ideal I R} = {{𝟬}, carrier R}) = field R"
  by (fast intro!: trivialideals_fieldI[OF carrnzero] field.all_ideals)


text ‹Like zeroprimeideal for domains›
lemma (in field) zeromaximalideal: "maximalideal {𝟬} R"
proof (intro maximalidealI zeroideal)
  from one_not_zero have "𝟭 ∉ {𝟬}" by simp
  with one_closed show "carrier R ≠ {𝟬}" by fast
next
  fix J
  assume Jideal: "ideal J R"
  then have "J ∈ {I. ideal I R}" by fast
  with all_ideals show "J = {𝟬} ∨ J = carrier R"
    by simp
qed

lemma (in cring) zeromaximalideal_fieldI:
  assumes zeromax: "maximalideal {𝟬} R"
  shows "field R"
proof (intro trivialideals_fieldI maximalideal.I_notcarr[OF zeromax])
  have "J = carrier R" if Jn0: "J ≠ {𝟬}" and idealJ: "ideal J R" for J
  proof -
    interpret ideal J R by (rule idealJ)
    have "{𝟬} ⊆ J"
      by force
    from zeromax idealJ this a_subset
    have "J = {𝟬} ∨ J = carrier R"
      by (rule maximalideal.I_maximal)
    with Jn0 show "J = carrier R"
      by simp
  qed
  then show "{I. ideal I R} = {{𝟬}, carrier R}"
    by (auto simp: zeroideal oneideal)
qed

lemma (in cring) zeromaximalideal_eq_field: "maximalideal {𝟬} R = field R"
  using field.zeromaximalideal zeromaximalideal_fieldI by blast

end