src/HOL/Algebra/Ideal.thy
 author haftmann Fri Apr 20 11:21:42 2007 +0200 (2007-04-20) changeset 22744 5cbe966d67a2 parent 20318 0e0ea63fe768 child 23350 50c5b0912a0c permissions -rw-r--r--
Isar definitions are now added explicitly to code theorem table
```     1 (*
```
```     2   Title:     HOL/Algebra/CIdeal.thy
```
```     3   Id:        \$Id\$
```
```     4   Author:    Stephan Hohe, TU Muenchen
```
```     5 *)
```
```     6
```
```     7 theory Ideal
```
```     8 imports Ring AbelCoset
```
```     9 begin
```
```    10
```
```    11 section {* Ideals *}
```
```    12
```
```    13 subsection {* General definition *}
```
```    14
```
```    15 locale ideal = additive_subgroup I R + ring R +
```
```    16   assumes I_l_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
```
```    17       and I_r_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
```
```    18
```
```    19 interpretation ideal \<subseteq> abelian_subgroup I R
```
```    20 apply (intro abelian_subgroupI3 abelian_group.intro)
```
```    21   apply (rule ideal.axioms, assumption)
```
```    22  apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, assumption)
```
```    23 apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, assumption)
```
```    24 done
```
```    25
```
```    26 lemma (in ideal) is_ideal:
```
```    27   "ideal I R"
```
```    28 by -
```
```    29
```
```    30 lemma idealI:
```
```    31   includes ring
```
```    32   assumes a_subgroup: "subgroup I \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
```
```    33       and I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
```
```    34       and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
```
```    35   shows "ideal I R"
```
```    36 by (intro ideal.intro ideal_axioms.intro additive_subgroupI, assumption+)
```
```    37
```
```    38
```
```    39 subsection {* Ideals Generated by a Subset of @{term [locale=ring] "carrier R"} *}
```
```    40
```
```    41 constdefs (structure R)
```
```    42   genideal :: "('a, 'b) ring_scheme \<Rightarrow> 'a set \<Rightarrow> 'a set"  ("Idl\<index> _" [80] 79)
```
```    43   "genideal R S \<equiv> Inter {I. ideal I R \<and> S \<subseteq> I}"
```
```    44
```
```    45
```
```    46 subsection {* Principal Ideals *}
```
```    47
```
```    48 locale principalideal = ideal +
```
```    49   assumes generate: "\<exists>i \<in> carrier R. I = Idl {i}"
```
```    50
```
```    51 lemma (in principalideal) is_principalideal:
```
```    52   shows "principalideal I R"
```
```    53 by -
```
```    54
```
```    55 lemma principalidealI:
```
```    56   includes ideal
```
```    57   assumes generate: "\<exists>i \<in> carrier R. I = Idl {i}"
```
```    58   shows "principalideal I R"
```
```    59 by (intro principalideal.intro principalideal_axioms.intro, assumption+)
```
```    60
```
```    61
```
```    62 subsection {* Maximal Ideals *}
```
```    63
```
```    64 locale maximalideal = ideal +
```
```    65   assumes I_notcarr: "carrier R \<noteq> I"
```
```    66       and I_maximal: "\<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
```
```    67
```
```    68 lemma (in maximalideal) is_maximalideal:
```
```    69  shows "maximalideal I R"
```
```    70 by -
```
```    71
```
```    72 lemma maximalidealI:
```
```    73   includes ideal
```
```    74   assumes I_notcarr: "carrier R \<noteq> I"
```
```    75      and I_maximal: "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
```
```    76   shows "maximalideal I R"
```
```    77 by (intro maximalideal.intro maximalideal_axioms.intro, assumption+)
```
```    78
```
```    79
```
```    80 subsection {* Prime Ideals *}
```
```    81
```
```    82 locale primeideal = ideal + cring +
```
```    83   assumes I_notcarr: "carrier R \<noteq> I"
```
```    84       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"
```
```    85
```
```    86 lemma (in primeideal) is_primeideal:
```
```    87  shows "primeideal I R"
```
```    88 by -
```
```    89
```
```    90 lemma primeidealI:
```
```    91   includes ideal
```
```    92   includes cring
```
```    93   assumes I_notcarr: "carrier R \<noteq> I"
```
```    94       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"
```
```    95   shows "primeideal I R"
```
```    96 by (intro primeideal.intro primeideal_axioms.intro, assumption+)
```
```    97
```
```    98 lemma primeidealI2:
```
```    99   includes additive_subgroup I R
```
```   100   includes cring
```
```   101   assumes I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
```
```   102       and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
```
```   103       and I_notcarr: "carrier R \<noteq> I"
```
```   104       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"
```
```   105   shows "primeideal I R"
```
```   106 apply (intro_locales)
```
```   107  apply (intro ideal_axioms.intro, assumption+)
```
```   108 apply (intro primeideal_axioms.intro, assumption+)
```
```   109 done
```
```   110
```
```   111
```
```   112 section {* Properties of Ideals *}
```
```   113
```
```   114 subsection {* Special Ideals *}
```
```   115
```
```   116 lemma (in ring) zeroideal:
```
```   117   shows "ideal {\<zero>} R"
```
```   118 apply (intro idealI subgroup.intro)
```
```   119       apply (rule is_ring)
```
```   120      apply simp+
```
```   121   apply (fold a_inv_def, simp)
```
```   122  apply simp+
```
```   123 done
```
```   124
```
```   125 lemma (in ring) oneideal:
```
```   126   shows "ideal (carrier R) R"
```
```   127 apply (intro idealI  subgroup.intro)
```
```   128       apply (rule is_ring)
```
```   129      apply simp+
```
```   130   apply (fold a_inv_def, simp)
```
```   131  apply simp+
```
```   132 done
```
```   133
```
```   134 lemma (in "domain") zeroprimeideal:
```
```   135  shows "primeideal {\<zero>} R"
```
```   136 apply (intro primeidealI)
```
```   137    apply (rule zeroideal)
```
```   138   apply (rule domain.axioms,assumption)
```
```   139  defer 1
```
```   140  apply (simp add: integral)
```
```   141 proof (rule ccontr, simp)
```
```   142   assume "carrier R = {\<zero>}"
```
```   143   from this have "\<one> = \<zero>" by (rule one_zeroI)
```
```   144   from this and one_not_zero
```
```   145       show "False" by simp
```
```   146 qed
```
```   147
```
```   148
```
```   149 subsection {* General Ideal Properies *}
```
```   150
```
```   151 lemma (in ideal) one_imp_carrier:
```
```   152   assumes I_one_closed: "\<one> \<in> I"
```
```   153   shows "I = carrier R"
```
```   154 apply (rule)
```
```   155 apply (rule)
```
```   156 apply (rule a_Hcarr, simp)
```
```   157 proof
```
```   158   fix x
```
```   159   assume xcarr: "x \<in> carrier R"
```
```   160   from I_one_closed and this
```
```   161       have "x \<otimes> \<one> \<in> I" by (intro I_l_closed)
```
```   162   from this and xcarr
```
```   163       show "x \<in> I" by simp
```
```   164 qed
```
```   165
```
```   166 lemma (in ideal) Icarr:
```
```   167   assumes iI: "i \<in> I"
```
```   168   shows "i \<in> carrier R"
```
```   169 by (rule a_Hcarr)
```
```   170
```
```   171
```
```   172 subsection {* Intersection of Ideals *}
```
```   173
```
```   174 text {* \paragraph{Intersection of two ideals} The intersection of any
```
```   175   two ideals is again an ideal in @{term R} *}
```
```   176 lemma (in ring) i_intersect:
```
```   177   includes ideal I R
```
```   178   includes ideal J R
```
```   179   shows "ideal (I \<inter> J) R"
```
```   180 apply (intro idealI subgroup.intro)
```
```   181       apply (rule is_ring)
```
```   182      apply (force simp add: a_subset)
```
```   183     apply (simp add: a_inv_def[symmetric])
```
```   184    apply simp
```
```   185   apply (simp add: a_inv_def[symmetric])
```
```   186  apply (clarsimp, rule)
```
```   187   apply (fast intro: ideal.I_l_closed ideal.intro prems)+
```
```   188 apply (clarsimp, rule)
```
```   189  apply (fast intro: ideal.I_r_closed ideal.intro prems)+
```
```   190 done
```
```   191
```
```   192
```
```   193 subsubsection {* Intersection of a Set of Ideals *}
```
```   194
```
```   195 text {* The intersection of any Number of Ideals is again
```
```   196         an Ideal in @{term R} *}
```
```   197 lemma (in ring) i_Intersect:
```
```   198   assumes Sideals: "\<And>I. I \<in> S \<Longrightarrow> ideal I R"
```
```   199     and notempty: "S \<noteq> {}"
```
```   200   shows "ideal (Inter S) R"
```
```   201 apply (unfold_locales)
```
```   202 apply (simp_all add: Inter_def INTER_def)
```
```   203       apply (rule, simp) defer 1
```
```   204       apply rule defer 1
```
```   205       apply rule defer 1
```
```   206       apply (fold a_inv_def, rule) defer 1
```
```   207       apply rule defer 1
```
```   208       apply rule defer 1
```
```   209 proof -
```
```   210   fix x
```
```   211   assume "\<forall>I\<in>S. x \<in> I"
```
```   212   hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   213
```
```   214   from notempty have "\<exists>I0. I0 \<in> S" by blast
```
```   215   from this obtain I0 where I0S: "I0 \<in> S" by auto
```
```   216
```
```   217   interpret ideal ["I0" "R"] by (rule Sideals[OF I0S])
```
```   218
```
```   219   from xI[OF I0S] have "x \<in> I0" .
```
```   220   from this and a_subset show "x \<in> carrier R" by fast
```
```   221 next
```
```   222   fix x y
```
```   223   assume "\<forall>I\<in>S. x \<in> I"
```
```   224   hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   225   assume "\<forall>I\<in>S. y \<in> I"
```
```   226   hence yI: "\<And>I. I \<in> S \<Longrightarrow> y \<in> I" by simp
```
```   227
```
```   228   fix J
```
```   229   assume JS: "J \<in> S"
```
```   230   interpret ideal ["J" "R"] by (rule Sideals[OF JS])
```
```   231   from xI[OF JS] and yI[OF JS]
```
```   232       show "x \<oplus> y \<in> J" by (rule a_closed)
```
```   233 next
```
```   234   fix J
```
```   235   assume JS: "J \<in> S"
```
```   236   interpret ideal ["J" "R"] by (rule Sideals[OF JS])
```
```   237   show "\<zero> \<in> J" by simp
```
```   238 next
```
```   239   fix x
```
```   240   assume "\<forall>I\<in>S. x \<in> I"
```
```   241   hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   242
```
```   243   fix J
```
```   244   assume JS: "J \<in> S"
```
```   245   interpret ideal ["J" "R"] by (rule Sideals[OF JS])
```
```   246
```
```   247   from xI[OF JS]
```
```   248       show "\<ominus> x \<in> J" by (rule a_inv_closed)
```
```   249 next
```
```   250   fix x y
```
```   251   assume "\<forall>I\<in>S. x \<in> I"
```
```   252   hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   253   assume ycarr: "y \<in> carrier R"
```
```   254
```
```   255   fix J
```
```   256   assume JS: "J \<in> S"
```
```   257   interpret ideal ["J" "R"] by (rule Sideals[OF JS])
```
```   258
```
```   259   from xI[OF JS] and ycarr
```
```   260       show "y \<otimes> x \<in> J" by (rule I_l_closed)
```
```   261 next
```
```   262   fix x y
```
```   263   assume "\<forall>I\<in>S. x \<in> I"
```
```   264   hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   265   assume ycarr: "y \<in> carrier R"
```
```   266
```
```   267   fix J
```
```   268   assume JS: "J \<in> S"
```
```   269   interpret ideal ["J" "R"] by (rule Sideals[OF JS])
```
```   270
```
```   271   from xI[OF JS] and ycarr
```
```   272       show "x \<otimes> y \<in> J" by (rule I_r_closed)
```
```   273 qed
```
```   274
```
```   275
```
```   276 subsection {* Addition of Ideals *}
```
```   277
```
```   278 lemma (in ring) add_ideals:
```
```   279   assumes idealI: "ideal I R"
```
```   280       and idealJ: "ideal J R"
```
```   281   shows "ideal (I <+> J) R"
```
```   282 apply (rule ideal.intro)
```
```   283   apply (rule add_additive_subgroups)
```
```   284    apply (intro ideal.axioms[OF idealI])
```
```   285   apply (intro ideal.axioms[OF idealJ])
```
```   286  apply assumption
```
```   287 apply (rule ideal_axioms.intro)
```
```   288  apply (simp add: set_add_defs, clarsimp) defer 1
```
```   289  apply (simp add: set_add_defs, clarsimp) defer 1
```
```   290 proof -
```
```   291   fix x i j
```
```   292   assume xcarr: "x \<in> carrier R"
```
```   293      and iI: "i \<in> I"
```
```   294      and jJ: "j \<in> J"
```
```   295   from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
```
```   296       have c: "(i \<oplus> j) \<otimes> x = (i \<otimes> x) \<oplus> (j \<otimes> x)" by algebra
```
```   297   from xcarr and iI
```
```   298       have a: "i \<otimes> x \<in> I" by (simp add: ideal.I_r_closed[OF idealI])
```
```   299   from xcarr and jJ
```
```   300       have b: "j \<otimes> x \<in> J" by (simp add: ideal.I_r_closed[OF idealJ])
```
```   301   from a b c
```
```   302       show "\<exists>ha\<in>I. \<exists>ka\<in>J. (i \<oplus> j) \<otimes> x = ha \<oplus> ka" by fast
```
```   303 next
```
```   304   fix x i j
```
```   305   assume xcarr: "x \<in> carrier R"
```
```   306      and iI: "i \<in> I"
```
```   307      and jJ: "j \<in> J"
```
```   308   from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
```
```   309       have c: "x \<otimes> (i \<oplus> j) = (x \<otimes> i) \<oplus> (x \<otimes> j)" by algebra
```
```   310   from xcarr and iI
```
```   311       have a: "x \<otimes> i \<in> I" by (simp add: ideal.I_l_closed[OF idealI])
```
```   312   from xcarr and jJ
```
```   313       have b: "x \<otimes> j \<in> J" by (simp add: ideal.I_l_closed[OF idealJ])
```
```   314   from a b c
```
```   315       show "\<exists>ha\<in>I. \<exists>ka\<in>J. x \<otimes> (i \<oplus> j) = ha \<oplus> ka" by fast
```
```   316 qed
```
```   317
```
```   318
```
```   319 subsection {* Ideals generated by a subset of @{term [locale=ring]
```
```   320   "carrier R"} *}
```
```   321
```
```   322 subsubsection {* Generation of Ideals in General Rings *}
```
```   323
```
```   324 text {* @{term genideal} generates an ideal *}
```
```   325 lemma (in ring) genideal_ideal:
```
```   326   assumes Scarr: "S \<subseteq> carrier R"
```
```   327   shows "ideal (Idl S) R"
```
```   328 unfolding genideal_def
```
```   329 proof (rule i_Intersect, fast, simp)
```
```   330   from oneideal and Scarr
```
```   331   show "\<exists>I. ideal I R \<and> S \<le> I" by fast
```
```   332 qed
```
```   333
```
```   334 lemma (in ring) genideal_self:
```
```   335   assumes "S \<subseteq> carrier R"
```
```   336   shows "S \<subseteq> Idl S"
```
```   337 unfolding genideal_def
```
```   338 by fast
```
```   339
```
```   340 lemma (in ring) genideal_self':
```
```   341   assumes carr: "i \<in> carrier R"
```
```   342   shows "i \<in> Idl {i}"
```
```   343 proof -
```
```   344   from carr
```
```   345       have "{i} \<subseteq> Idl {i}" by (fast intro!: genideal_self)
```
```   346   thus "i \<in> Idl {i}" by fast
```
```   347 qed
```
```   348
```
```   349 text {* @{term genideal} generates the minimal ideal *}
```
```   350 lemma (in ring) genideal_minimal:
```
```   351   assumes a: "ideal I R"
```
```   352       and b: "S \<subseteq> I"
```
```   353   shows "Idl S \<subseteq> I"
```
```   354 unfolding genideal_def
```
```   355 by (rule, elim InterD, simp add: a b)
```
```   356
```
```   357 text {* Generated ideals and subsets *}
```
```   358 lemma (in ring) Idl_subset_ideal:
```
```   359   assumes Iideal: "ideal I R"
```
```   360       and Hcarr: "H \<subseteq> carrier R"
```
```   361   shows "(Idl H \<subseteq> I) = (H \<subseteq> I)"
```
```   362 proof
```
```   363   assume a: "Idl H \<subseteq> I"
```
```   364   have "H \<subseteq> Idl H" by (rule genideal_self)
```
```   365   from this and a
```
```   366       show "H \<subseteq> I" by simp
```
```   367 next
```
```   368   fix x
```
```   369   assume HI: "H \<subseteq> I"
```
```   370
```
```   371   from Iideal and HI
```
```   372       have "I \<in> {I. ideal I R \<and> H \<subseteq> I}" by fast
```
```   373   from this
```
```   374       show "Idl H \<subseteq> I"
```
```   375       unfolding genideal_def
```
```   376       by fast
```
```   377 qed
```
```   378
```
```   379 lemma (in ring) subset_Idl_subset:
```
```   380   assumes Icarr: "I \<subseteq> carrier R"
```
```   381       and HI: "H \<subseteq> I"
```
```   382   shows "Idl H \<subseteq> Idl I"
```
```   383 proof -
```
```   384   from HI and genideal_self[OF Icarr]
```
```   385       have HIdlI: "H \<subseteq> Idl I" by fast
```
```   386
```
```   387   from Icarr
```
```   388       have Iideal: "ideal (Idl I) R" by (rule genideal_ideal)
```
```   389   from HI and Icarr
```
```   390       have "H \<subseteq> carrier R" by fast
```
```   391   from Iideal and this
```
```   392       have "(H \<subseteq> Idl I) = (Idl H \<subseteq> Idl I)"
```
```   393       by (rule Idl_subset_ideal[symmetric])
```
```   394
```
```   395   from HIdlI and this
```
```   396       show "Idl H \<subseteq> Idl I" by simp
```
```   397 qed
```
```   398
```
```   399 lemma (in ring) Idl_subset_ideal':
```
```   400   assumes acarr: "a \<in> carrier R" and bcarr: "b \<in> carrier R"
```
```   401   shows "(Idl {a} \<subseteq> Idl {b}) = (a \<in> Idl {b})"
```
```   402 apply (subst Idl_subset_ideal[OF genideal_ideal[of "{b}"], of "{a}"])
```
```   403   apply (fast intro: bcarr, fast intro: acarr)
```
```   404 apply fast
```
```   405 done
```
```   406
```
```   407 lemma (in ring) genideal_zero:
```
```   408   "Idl {\<zero>} = {\<zero>}"
```
```   409 apply rule
```
```   410  apply (rule genideal_minimal[OF zeroideal], simp)
```
```   411 apply (simp add: genideal_self')
```
```   412 done
```
```   413
```
```   414 lemma (in ring) genideal_one:
```
```   415   "Idl {\<one>} = carrier R"
```
```   416 proof -
```
```   417   interpret ideal ["Idl {\<one>}" "R"] by (rule genideal_ideal, fast intro: one_closed)
```
```   418   show "Idl {\<one>} = carrier R"
```
```   419   apply (rule, rule a_subset)
```
```   420   apply (simp add: one_imp_carrier genideal_self')
```
```   421   done
```
```   422 qed
```
```   423
```
```   424
```
```   425 subsubsection {* Generation of Principal Ideals in Commutative Rings *}
```
```   426
```
```   427 constdefs (structure R)
```
```   428   cgenideal :: "('a, 'b) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a set"  ("PIdl\<index> _" [80] 79)
```
```   429   "cgenideal R a \<equiv> { x \<otimes> a | x. x \<in> carrier R }"
```
```   430
```
```   431 text {* genhideal (?) really generates an ideal *}
```
```   432 lemma (in cring) cgenideal_ideal:
```
```   433   assumes acarr: "a \<in> carrier R"
```
```   434   shows "ideal (PIdl a) R"
```
```   435 apply (unfold cgenideal_def)
```
```   436 apply (rule idealI[OF is_ring])
```
```   437    apply (rule subgroup.intro)
```
```   438       apply (simp_all add: monoid_record_simps)
```
```   439       apply (blast intro: acarr m_closed)
```
```   440       apply clarsimp defer 1
```
```   441       defer 1
```
```   442       apply (fold a_inv_def, clarsimp) defer 1
```
```   443       apply clarsimp defer 1
```
```   444       apply clarsimp defer 1
```
```   445 proof -
```
```   446   fix x y
```
```   447   assume xcarr: "x \<in> carrier R"
```
```   448      and ycarr: "y \<in> carrier R"
```
```   449   note carr = acarr xcarr ycarr
```
```   450
```
```   451   from carr
```
```   452       have "x \<otimes> a \<oplus> y \<otimes> a = (x \<oplus> y) \<otimes> a" by (simp add: l_distr)
```
```   453   from this and carr
```
```   454       show "\<exists>z. x \<otimes> a \<oplus> y \<otimes> a = z \<otimes> a \<and> z \<in> carrier R" by fast
```
```   455 next
```
```   456   from l_null[OF acarr, symmetric] and zero_closed
```
```   457       show "\<exists>x. \<zero> = x \<otimes> a \<and> x \<in> carrier R" by fast
```
```   458 next
```
```   459   fix x
```
```   460   assume xcarr: "x \<in> carrier R"
```
```   461   note carr = acarr xcarr
```
```   462
```
```   463   from carr
```
```   464       have "\<ominus> (x \<otimes> a) = (\<ominus> x) \<otimes> a" by (simp add: l_minus)
```
```   465   from this and carr
```
```   466       show "\<exists>z. \<ominus> (x \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R" by fast
```
```   467 next
```
```   468   fix x y
```
```   469   assume xcarr: "x \<in> carrier R"
```
```   470      and ycarr: "y \<in> carrier R"
```
```   471   note carr = acarr xcarr ycarr
```
```   472
```
```   473   from carr
```
```   474       have "y \<otimes> a \<otimes> x = (y \<otimes> x) \<otimes> a" by (simp add: m_assoc, simp add: m_comm)
```
```   475   from this and carr
```
```   476       show "\<exists>z. y \<otimes> a \<otimes> x = z \<otimes> a \<and> z \<in> carrier R" by fast
```
```   477 next
```
```   478   fix x y
```
```   479   assume xcarr: "x \<in> carrier R"
```
```   480      and ycarr: "y \<in> carrier R"
```
```   481   note carr = acarr xcarr ycarr
```
```   482
```
```   483   from carr
```
```   484       have "x \<otimes> (y \<otimes> a) = (x \<otimes> y) \<otimes> a" by (simp add: m_assoc)
```
```   485   from this and carr
```
```   486       show "\<exists>z. x \<otimes> (y \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R" by fast
```
```   487 qed
```
```   488
```
```   489 lemma (in ring) cgenideal_self:
```
```   490   assumes icarr: "i \<in> carrier R"
```
```   491   shows "i \<in> PIdl i"
```
```   492 unfolding cgenideal_def
```
```   493 proof simp
```
```   494   from icarr
```
```   495       have "i = \<one> \<otimes> i" by simp
```
```   496   from this and icarr
```
```   497       show "\<exists>x. i = x \<otimes> i \<and> x \<in> carrier R" by fast
```
```   498 qed
```
```   499
```
```   500 text {* @{const "cgenideal"} is minimal *}
```
```   501
```
```   502 lemma (in ring) cgenideal_minimal:
```
```   503   includes ideal J R
```
```   504   assumes aJ: "a \<in> J"
```
```   505   shows "PIdl a \<subseteq> J"
```
```   506 unfolding cgenideal_def
```
```   507 by (rule, clarify, rule I_l_closed)
```
```   508
```
```   509
```
```   510 lemma (in cring) cgenideal_eq_genideal:
```
```   511   assumes icarr: "i \<in> carrier R"
```
```   512   shows "PIdl i = Idl {i}"
```
```   513 apply rule
```
```   514  apply (intro cgenideal_minimal)
```
```   515   apply (rule genideal_ideal, fast intro: icarr)
```
```   516  apply (rule genideal_self', fast intro: icarr)
```
```   517 apply (intro genideal_minimal)
```
```   518  apply (rule cgenideal_ideal, assumption)
```
```   519 apply (simp, rule cgenideal_self, assumption)
```
```   520 done
```
```   521
```
```   522 lemma (in cring) cgenideal_eq_rcos:
```
```   523  "PIdl i = carrier R #> i"
```
```   524 unfolding cgenideal_def r_coset_def
```
```   525 by fast
```
```   526
```
```   527 lemma (in cring) cgenideal_is_principalideal:
```
```   528   assumes icarr: "i \<in> carrier R"
```
```   529   shows "principalideal (PIdl i) R"
```
```   530 apply (rule principalidealI)
```
```   531 apply (rule cgenideal_ideal, assumption)
```
```   532 proof -
```
```   533   from icarr
```
```   534       have "PIdl i = Idl {i}" by (rule cgenideal_eq_genideal)
```
```   535   from icarr and this
```
```   536       show "\<exists>i'\<in>carrier R. PIdl i = Idl {i'}" by fast
```
```   537 qed
```
```   538
```
```   539
```
```   540 subsection {* Union of Ideals *}
```
```   541
```
```   542 lemma (in ring) union_genideal:
```
```   543   assumes idealI: "ideal I R"
```
```   544       and idealJ: "ideal J R"
```
```   545   shows "Idl (I \<union> J) = I <+> J"
```
```   546 apply rule
```
```   547  apply (rule ring.genideal_minimal)
```
```   548    apply assumption
```
```   549   apply (rule add_ideals[OF idealI idealJ])
```
```   550  apply (rule)
```
```   551  apply (simp add: set_add_defs) apply (elim disjE) defer 1 defer 1
```
```   552  apply (rule) apply (simp add: set_add_defs genideal_def) apply clarsimp defer 1
```
```   553 proof -
```
```   554   fix x
```
```   555   assume xI: "x \<in> I"
```
```   556   have ZJ: "\<zero> \<in> J"
```
```   557       by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealJ])
```
```   558   from ideal.Icarr[OF idealI xI]
```
```   559       have "x = x \<oplus> \<zero>" by algebra
```
```   560   from xI and ZJ and this
```
```   561       show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k" by fast
```
```   562 next
```
```   563   fix x
```
```   564   assume xJ: "x \<in> J"
```
```   565   have ZI: "\<zero> \<in> I"
```
```   566       by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealI])
```
```   567   from ideal.Icarr[OF idealJ xJ]
```
```   568       have "x = \<zero> \<oplus> x" by algebra
```
```   569   from ZI and xJ and this
```
```   570       show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k" by fast
```
```   571 next
```
```   572   fix i j K
```
```   573   assume iI: "i \<in> I"
```
```   574      and jJ: "j \<in> J"
```
```   575      and idealK: "ideal K R"
```
```   576      and IK: "I \<subseteq> K"
```
```   577      and JK: "J \<subseteq> K"
```
```   578   from iI and IK
```
```   579      have iK: "i \<in> K" by fast
```
```   580   from jJ and JK
```
```   581      have jK: "j \<in> K" by fast
```
```   582   from iK and jK
```
```   583      show "i \<oplus> j \<in> K" by (intro additive_subgroup.a_closed) (rule ideal.axioms[OF idealK])
```
```   584 qed
```
```   585
```
```   586
```
```   587 subsection {* Properties of Principal Ideals *}
```
```   588
```
```   589 text {* @{text "\<zero>"} generates the zero ideal *}
```
```   590 lemma (in ring) zero_genideal:
```
```   591   shows "Idl {\<zero>} = {\<zero>}"
```
```   592 apply rule
```
```   593 apply (simp add: genideal_minimal zeroideal)
```
```   594 apply (fast intro!: genideal_self)
```
```   595 done
```
```   596
```
```   597 text {* @{text "\<one>"} generates the unit ideal *}
```
```   598 lemma (in ring) one_genideal:
```
```   599   shows "Idl {\<one>} = carrier R"
```
```   600 proof -
```
```   601   have "\<one> \<in> Idl {\<one>}" by (simp add: genideal_self')
```
```   602   thus "Idl {\<one>} = carrier R" by (intro ideal.one_imp_carrier, fast intro: genideal_ideal)
```
```   603 qed
```
```   604
```
```   605
```
```   606 text {* The zero ideal is a principal ideal *}
```
```   607 corollary (in ring) zeropideal:
```
```   608   shows "principalideal {\<zero>} R"
```
```   609 apply (rule principalidealI)
```
```   610  apply (rule zeroideal)
```
```   611 apply (blast intro!: zero_closed zero_genideal[symmetric])
```
```   612 done
```
```   613
```
```   614 text {* The unit ideal is a principal ideal *}
```
```   615 corollary (in ring) onepideal:
```
```   616   shows "principalideal (carrier R) R"
```
```   617 apply (rule principalidealI)
```
```   618  apply (rule oneideal)
```
```   619 apply (blast intro!: one_closed one_genideal[symmetric])
```
```   620 done
```
```   621
```
```   622
```
```   623 text {* Every principal ideal is a right coset of the carrier *}
```
```   624 lemma (in principalideal) rcos_generate:
```
```   625   includes cring
```
```   626   shows "\<exists>x\<in>I. I = carrier R #> x"
```
```   627 proof -
```
```   628   from generate
```
```   629       obtain i
```
```   630         where icarr: "i \<in> carrier R"
```
```   631         and I1: "I = Idl {i}"
```
```   632       by fast+
```
```   633
```
```   634   from icarr and genideal_self[of "{i}"]
```
```   635       have "i \<in> Idl {i}" by fast
```
```   636   hence iI: "i \<in> I" by (simp add: I1)
```
```   637
```
```   638   from I1 icarr
```
```   639       have I2: "I = PIdl i" by (simp add: cgenideal_eq_genideal)
```
```   640
```
```   641   have "PIdl i = carrier R #> i"
```
```   642       unfolding cgenideal_def r_coset_def
```
```   643       by fast
```
```   644
```
```   645   from I2 and this
```
```   646       have "I = carrier R #> i" by simp
```
```   647
```
```   648   from iI and this
```
```   649       show "\<exists>x\<in>I. I = carrier R #> x" by fast
```
```   650 qed
```
```   651
```
```   652
```
```   653 subsection {* Prime Ideals *}
```
```   654
```
```   655 lemma (in ideal) primeidealCD:
```
```   656   includes cring
```
```   657   assumes notprime: "\<not> primeideal I R"
```
```   658   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)"
```
```   659 proof (rule ccontr, clarsimp)
```
```   660   assume InR: "carrier R \<noteq> I"
```
```   661      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)"
```
```   662   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
```
```   663   have "primeideal I R"
```
```   664       apply (rule primeideal.intro, assumption+)
```
```   665       by (rule primeideal_axioms.intro, rule InR, erule I_prime)
```
```   666   from this and notprime
```
```   667       show "False" by simp
```
```   668 qed
```
```   669
```
```   670 lemma (in ideal) primeidealCE:
```
```   671   includes cring
```
```   672   assumes notprime: "\<not> primeideal I R"
```
```   673     and elim1: "carrier R = I \<Longrightarrow> P"
```
```   674     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"
```
```   675   shows "P"
```
```   676 proof -
```
```   677   from notprime
```
```   678   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)"
```
```   679       by (intro primeidealCD)
```
```   680   thus "P"
```
```   681       apply (rule disjE)
```
```   682       by (erule elim1, erule elim2)
```
```   683 qed
```
```   684
```
```   685 text {* If @{text "{\<zero>}"} is a prime ideal of a commutative ring, the ring is a domain *}
```
```   686 lemma (in cring) zeroprimeideal_domainI:
```
```   687   assumes pi: "primeideal {\<zero>} R"
```
```   688   shows "domain R"
```
```   689 apply (rule domain.intro, assumption+)
```
```   690 apply (rule domain_axioms.intro)
```
```   691 proof (rule ccontr, simp)
```
```   692   interpret primeideal ["{\<zero>}" "R"] by (rule pi)
```
```   693   assume "\<one> = \<zero>"
```
```   694   hence "carrier R = {\<zero>}" by (rule one_zeroD)
```
```   695   from this[symmetric] and I_notcarr
```
```   696       show "False" by simp
```
```   697 next
```
```   698   interpret primeideal ["{\<zero>}" "R"] by (rule pi)
```
```   699   fix a b
```
```   700   assume ab: "a \<otimes> b = \<zero>"
```
```   701      and carr: "a \<in> carrier R" "b \<in> carrier R"
```
```   702   from ab
```
```   703       have abI: "a \<otimes> b \<in> {\<zero>}" by fast
```
```   704   from carr and this
```
```   705       have "a \<in> {\<zero>} \<or> b \<in> {\<zero>}" by (rule I_prime)
```
```   706   thus "a = \<zero> \<or> b = \<zero>" by simp
```
```   707 qed
```
```   708
```
```   709 corollary (in cring) domain_eq_zeroprimeideal:
```
```   710   shows "domain R = primeideal {\<zero>} R"
```
```   711 apply rule
```
```   712  apply (erule domain.zeroprimeideal)
```
```   713 apply (erule zeroprimeideal_domainI)
```
```   714 done
```
```   715
```
```   716
```
```   717 subsection {* Maximal Ideals *}
```
```   718
```
```   719 lemma (in ideal) helper_I_closed:
```
```   720   assumes carr: "a \<in> carrier R" "x \<in> carrier R" "y \<in> carrier R"
```
```   721       and axI: "a \<otimes> x \<in> I"
```
```   722   shows "a \<otimes> (x \<otimes> y) \<in> I"
```
```   723 proof -
```
```   724   from axI and carr
```
```   725      have "(a \<otimes> x) \<otimes> y \<in> I" by (simp add: I_r_closed)
```
```   726   also from carr
```
```   727      have "(a \<otimes> x) \<otimes> y = a \<otimes> (x \<otimes> y)" by (simp add: m_assoc)
```
```   728   finally
```
```   729      show "a \<otimes> (x \<otimes> y) \<in> I" .
```
```   730 qed
```
```   731
```
```   732 lemma (in ideal) helper_max_prime:
```
```   733   includes cring
```
```   734   assumes acarr: "a \<in> carrier R"
```
```   735   shows "ideal {x\<in>carrier R. a \<otimes> x \<in> I} R"
```
```   736 apply (rule idealI)
```
```   737    apply (rule cring.axioms[OF is_cring])
```
```   738   apply (rule subgroup.intro)
```
```   739      apply (simp, fast)
```
```   740     apply clarsimp apply (simp add: r_distr acarr)
```
```   741    apply (simp add: acarr)
```
```   742   apply (simp add: a_inv_def[symmetric], clarify) defer 1
```
```   743   apply clarsimp defer 1
```
```   744   apply (fast intro!: helper_I_closed acarr)
```
```   745 proof -
```
```   746   fix x
```
```   747   assume xcarr: "x \<in> carrier R"
```
```   748      and ax: "a \<otimes> x \<in> I"
```
```   749   from ax and acarr xcarr
```
```   750       have "\<ominus>(a \<otimes> x) \<in> I" by simp
```
```   751   also from acarr xcarr
```
```   752       have "\<ominus>(a \<otimes> x) = a \<otimes> (\<ominus>x)" by algebra
```
```   753   finally
```
```   754       show "a \<otimes> (\<ominus>x) \<in> I" .
```
```   755   from acarr
```
```   756       have "a \<otimes> \<zero> = \<zero>" by simp
```
```   757 next
```
```   758   fix x y
```
```   759   assume xcarr: "x \<in> carrier R"
```
```   760      and ycarr: "y \<in> carrier R"
```
```   761      and ayI: "a \<otimes> y \<in> I"
```
```   762   from ayI and acarr xcarr ycarr
```
```   763       have "a \<otimes> (y \<otimes> x) \<in> I" by (simp add: helper_I_closed)
```
```   764   moreover from xcarr ycarr
```
```   765       have "y \<otimes> x = x \<otimes> y" by (simp add: m_comm)
```
```   766   ultimately
```
```   767       show "a \<otimes> (x \<otimes> y) \<in> I" by simp
```
```   768 qed
```
```   769
```
```   770 text {* In a cring every maximal ideal is prime *}
```
```   771 lemma (in cring) maximalideal_is_prime:
```
```   772   includes maximalideal
```
```   773   shows "primeideal I R"
```
```   774 apply (rule ccontr)
```
```   775 apply (rule primeidealCE)
```
```   776   apply assumption+
```
```   777  apply (simp add: I_notcarr)
```
```   778 proof -
```
```   779   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"
```
```   780   from this
```
```   781       obtain a b
```
```   782         where acarr: "a \<in> carrier R"
```
```   783         and bcarr: "b \<in> carrier R"
```
```   784         and abI: "a \<otimes> b \<in> I"
```
```   785         and anI: "a \<notin> I"
```
```   786         and bnI: "b \<notin> I"
```
```   787       by fast
```
```   788   def J \<equiv> "{x\<in>carrier R. a \<otimes> x \<in> I}"
```
```   789
```
```   790   from acarr
```
```   791       have idealJ: "ideal J R" by (unfold J_def, intro helper_max_prime)
```
```   792
```
```   793   have IsubJ: "I \<subseteq> J"
```
```   794   proof
```
```   795     fix x
```
```   796     assume xI: "x \<in> I"
```
```   797     from this and acarr
```
```   798         have "a \<otimes> x \<in> I" by (intro I_l_closed)
```
```   799     from xI[THEN a_Hcarr] this
```
```   800         show "x \<in> J" by (unfold J_def, fast)
```
```   801   qed
```
```   802
```
```   803   from abI and acarr bcarr
```
```   804       have "b \<in> J" by (unfold J_def, fast)
```
```   805   from bnI and this
```
```   806       have JnI: "J \<noteq> I" by fast
```
```   807   from acarr
```
```   808       have "a = a \<otimes> \<one>" by algebra
```
```   809   from this and anI
```
```   810       have "a \<otimes> \<one> \<notin> I" by simp
```
```   811   from one_closed and this
```
```   812       have "\<one> \<notin> J" by (unfold J_def, fast)
```
```   813   hence Jncarr: "J \<noteq> carrier R" by fast
```
```   814
```
```   815   interpret ideal ["J" "R"] by (rule idealJ)
```
```   816
```
```   817   have "J = I \<or> J = carrier R"
```
```   818      apply (intro I_maximal)
```
```   819      apply (rule idealJ)
```
```   820      apply (rule IsubJ)
```
```   821      apply (rule a_subset)
```
```   822      done
```
```   823
```
```   824   from this and JnI and Jncarr
```
```   825       show "False" by simp
```
```   826 qed
```
```   827
```
```   828
```
```   829 subsection {* Derived Theorems Involving Ideals *}
```
```   830
```
```   831 --"A non-zero cring that has only the two trivial ideals is a field"
```
```   832 lemma (in cring) trivialideals_fieldI:
```
```   833   assumes carrnzero: "carrier R \<noteq> {\<zero>}"
```
```   834       and haveideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
```
```   835   shows "field R"
```
```   836 apply (rule cring_fieldI)
```
```   837 apply (rule, rule, rule)
```
```   838  apply (erule Units_closed)
```
```   839 defer 1
```
```   840   apply rule
```
```   841 defer 1
```
```   842 proof (rule ccontr, simp)
```
```   843   assume zUnit: "\<zero> \<in> Units R"
```
```   844   hence a: "\<zero> \<otimes> inv \<zero> = \<one>" by (rule Units_r_inv)
```
```   845   from zUnit
```
```   846       have "\<zero> \<otimes> inv \<zero> = \<zero>" by (intro l_null, rule Units_inv_closed)
```
```   847   from a[symmetric] and this
```
```   848       have "\<one> = \<zero>" by simp
```
```   849   hence "carrier R = {\<zero>}" by (rule one_zeroD)
```
```   850   from this and carrnzero
```
```   851       show "False" by simp
```
```   852 next
```
```   853   fix x
```
```   854   assume xcarr': "x \<in> carrier R - {\<zero>}"
```
```   855   hence xcarr: "x \<in> carrier R" by fast
```
```   856   from xcarr'
```
```   857       have xnZ: "x \<noteq> \<zero>" by fast
```
```   858   from xcarr
```
```   859       have xIdl: "ideal (PIdl x) R" by (intro cgenideal_ideal, fast)
```
```   860
```
```   861   from xcarr
```
```   862       have "x \<in> PIdl x" by (intro cgenideal_self, fast)
```
```   863   from this and xnZ
```
```   864       have "PIdl x \<noteq> {\<zero>}" by fast
```
```   865   from haveideals and this
```
```   866       have "PIdl x = carrier R"
```
```   867       by (blast intro!: xIdl)
```
```   868   hence "\<one> \<in> PIdl x" by simp
```
```   869   hence "\<exists>y. \<one> = y \<otimes> x \<and> y \<in> carrier R" unfolding cgenideal_def by blast
```
```   870   from this
```
```   871       obtain y
```
```   872         where ycarr: " y \<in> carrier R"
```
```   873         and ylinv: "\<one> = y \<otimes> x"
```
```   874       by fast+
```
```   875   from ylinv and xcarr ycarr
```
```   876       have yrinv: "\<one> = x \<otimes> y" by (simp add: m_comm)
```
```   877   from ycarr and ylinv[symmetric] and yrinv[symmetric]
```
```   878       have "\<exists>y \<in> carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
```
```   879   from this and xcarr
```
```   880       show "x \<in> Units R"
```
```   881       unfolding Units_def
```
```   882       by fast
```
```   883 qed
```
```   884
```
```   885 lemma (in field) all_ideals:
```
```   886   shows "{I. ideal I R} = {{\<zero>}, carrier R}"
```
```   887 apply (rule, rule)
```
```   888 proof -
```
```   889   fix I
```
```   890   assume a: "I \<in> {I. ideal I R}"
```
```   891   with this
```
```   892       interpret ideal ["I" "R"] by simp
```
```   893
```
```   894   show "I \<in> {{\<zero>}, carrier R}"
```
```   895   proof (cases "\<exists>a. a \<in> I - {\<zero>}")
```
```   896     assume "\<exists>a. a \<in> I - {\<zero>}"
```
```   897     from this
```
```   898         obtain a
```
```   899           where aI: "a \<in> I"
```
```   900           and anZ: "a \<noteq> \<zero>"
```
```   901         by fast+
```
```   902     from aI[THEN a_Hcarr] anZ
```
```   903         have aUnit: "a \<in> Units R" by (simp add: field_Units)
```
```   904     hence a: "a \<otimes> inv a = \<one>" by (rule Units_r_inv)
```
```   905     from aI and aUnit
```
```   906         have "a \<otimes> inv a \<in> I" by (simp add: I_r_closed)
```
```   907     hence oneI: "\<one> \<in> I" by (simp add: a[symmetric])
```
```   908
```
```   909     have "carrier R \<subseteq> I"
```
```   910     proof
```
```   911       fix x
```
```   912       assume xcarr: "x \<in> carrier R"
```
```   913       from oneI and this
```
```   914           have "\<one> \<otimes> x \<in> I" by (rule I_r_closed)
```
```   915       from this and xcarr
```
```   916           show "x \<in> I" by simp
```
```   917     qed
```
```   918     from this and a_subset
```
```   919         have "I = carrier R" by fast
```
```   920     thus "I \<in> {{\<zero>}, carrier R}" by fast
```
```   921   next
```
```   922     assume "\<not> (\<exists>a. a \<in> I - {\<zero>})"
```
```   923     hence IZ: "\<And>a. a \<in> I \<Longrightarrow> a = \<zero>" by simp
```
```   924
```
```   925     have a: "I \<subseteq> {\<zero>}"
```
```   926     proof
```
```   927       fix x
```
```   928       assume "x \<in> I"
```
```   929       hence "x = \<zero>" by (rule IZ)
```
```   930       thus "x \<in> {\<zero>}" by fast
```
```   931     qed
```
```   932
```
```   933     have "\<zero> \<in> I" by simp
```
```   934     hence "{\<zero>} \<subseteq> I" by fast
```
```   935
```
```   936     from this and a
```
```   937         have "I = {\<zero>}" by fast
```
```   938     thus "I \<in> {{\<zero>}, carrier R}" by fast
```
```   939   qed
```
```   940 qed (simp add: zeroideal oneideal)
```
```   941
```
```   942 --"Jacobson Theorem 2.2"
```
```   943 lemma (in cring) trivialideals_eq_field:
```
```   944   assumes carrnzero: "carrier R \<noteq> {\<zero>}"
```
```   945   shows "({I. ideal I R} = {{\<zero>}, carrier R}) = field R"
```
```   946 by (fast intro!: trivialideals_fieldI[OF carrnzero] field.all_ideals)
```
```   947
```
```   948
```
```   949 text {* Like zeroprimeideal for domains *}
```
```   950 lemma (in field) zeromaximalideal:
```
```   951   "maximalideal {\<zero>} R"
```
```   952 apply (rule maximalidealI)
```
```   953   apply (rule zeroideal)
```
```   954 proof-
```
```   955   from one_not_zero
```
```   956       have "\<one> \<notin> {\<zero>}" by simp
```
```   957   from this and one_closed
```
```   958       show "carrier R \<noteq> {\<zero>}" by fast
```
```   959 next
```
```   960   fix J
```
```   961   assume Jideal: "ideal J R"
```
```   962   hence "J \<in> {I. ideal I R}"
```
```   963       by fast
```
```   964
```
```   965   from this and all_ideals
```
```   966       show "J = {\<zero>} \<or> J = carrier R" by simp
```
```   967 qed
```
```   968
```
```   969 lemma (in cring) zeromaximalideal_fieldI:
```
```   970   assumes zeromax: "maximalideal {\<zero>} R"
```
```   971   shows "field R"
```
```   972 apply (rule trivialideals_fieldI, rule maximalideal.I_notcarr[OF zeromax])
```
```   973 apply rule apply clarsimp defer 1
```
```   974  apply (simp add: zeroideal oneideal)
```
```   975 proof -
```
```   976   fix J
```
```   977   assume Jn0: "J \<noteq> {\<zero>}"
```
```   978      and idealJ: "ideal J R"
```
```   979   interpret ideal ["J" "R"] by (rule idealJ)
```
```   980   have "{\<zero>} \<subseteq> J" by (rule ccontr, simp)
```
```   981   from zeromax and idealJ and this and a_subset
```
```   982       have "J = {\<zero>} \<or> J = carrier R" by (rule maximalideal.I_maximal)
```
```   983   from this and Jn0
```
```   984       show "J = carrier R" by simp
```
```   985 qed
```
```   986
```
```   987 lemma (in cring) zeromaximalideal_eq_field:
```
```   988   "maximalideal {\<zero>} R = field R"
```
```   989 apply rule
```
```   990  apply (erule zeromaximalideal_fieldI)
```
```   991 apply (erule field.zeromaximalideal)
```
```   992 done
```
```   993
```
```   994 end
```