(*
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 fact
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 fact
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 fact
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 fact
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"
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:
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"
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:
includes ideal J R
assumes aJ: "a \<in> J"
shows "PIdl a \<subseteq> J"
unfolding cgenideal_def
apply rule
apply clarify
using aJ
apply (erule I_l_closed)
done
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"
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"
using primeidealCD [OF R.is_cring notprime] by blast
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 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
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