src/HOL/Algebra/Ideal.thy
 author wenzelm Sat Oct 10 16:26:23 2015 +0200 (2015-10-10) changeset 61382 efac889fccbc parent 47409 c5be1120980d child 61506 436b7fe89cdc permissions -rw-r--r--
isabelle update_cartouches;
```     1 (*  Title:      HOL/Algebra/Ideal.thy
```
```     2     Author:     Stephan Hohe, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 theory Ideal
```
```     6 imports Ring AbelCoset
```
```     7 begin
```
```     8
```
```     9 section \<open>Ideals\<close>
```
```    10
```
```    11 subsection \<open>Definitions\<close>
```
```    12
```
```    13 subsubsection \<open>General definition\<close>
```
```    14
```
```    15 locale ideal = additive_subgroup I R + ring R for I and R (structure) +
```
```    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 sublocale ideal \<subseteq> abelian_subgroup I R
```
```    20   apply (intro abelian_subgroupI3 abelian_group.intro)
```
```    21     apply (rule ideal.axioms, rule ideal_axioms)
```
```    22    apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)
```
```    23   apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)
```
```    24   done
```
```    25
```
```    26 lemma (in ideal) is_ideal: "ideal I R"
```
```    27   by (rule ideal_axioms)
```
```    28
```
```    29 lemma idealI:
```
```    30   fixes R (structure)
```
```    31   assumes "ring R"
```
```    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 proof -
```
```    37   interpret ring R by fact
```
```    38   show ?thesis  apply (intro ideal.intro ideal_axioms.intro additive_subgroupI)
```
```    39      apply (rule a_subgroup)
```
```    40     apply (rule is_ring)
```
```    41    apply (erule (1) I_l_closed)
```
```    42   apply (erule (1) I_r_closed)
```
```    43   done
```
```    44 qed
```
```    45
```
```    46
```
```    47 subsubsection (in ring) \<open>Ideals Generated by a Subset of @{term "carrier R"}\<close>
```
```    48
```
```    49 definition genideal :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set"  ("Idl\<index> _"  79)
```
```    50   where "genideal R S = Inter {I. ideal I R \<and> S \<subseteq> I}"
```
```    51
```
```    52 subsubsection \<open>Principal Ideals\<close>
```
```    53
```
```    54 locale principalideal = ideal +
```
```    55   assumes generate: "\<exists>i \<in> carrier R. I = Idl {i}"
```
```    56
```
```    57 lemma (in principalideal) is_principalideal: "principalideal I R"
```
```    58   by (rule principalideal_axioms)
```
```    59
```
```    60 lemma principalidealI:
```
```    61   fixes R (structure)
```
```    62   assumes "ideal I R"
```
```    63     and generate: "\<exists>i \<in> carrier R. I = Idl {i}"
```
```    64   shows "principalideal I R"
```
```    65 proof -
```
```    66   interpret ideal I R by fact
```
```    67   show ?thesis
```
```    68     by (intro principalideal.intro principalideal_axioms.intro)
```
```    69       (rule is_ideal, rule generate)
```
```    70 qed
```
```    71
```
```    72
```
```    73 subsubsection \<open>Maximal Ideals\<close>
```
```    74
```
```    75 locale maximalideal = ideal +
```
```    76   assumes I_notcarr: "carrier R \<noteq> I"
```
```    77     and I_maximal: "\<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
```
```    78
```
```    79 lemma (in maximalideal) is_maximalideal: "maximalideal I R"
```
```    80   by (rule maximalideal_axioms)
```
```    81
```
```    82 lemma maximalidealI:
```
```    83   fixes R
```
```    84   assumes "ideal I R"
```
```    85     and I_notcarr: "carrier R \<noteq> I"
```
```    86     and I_maximal: "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
```
```    87   shows "maximalideal I R"
```
```    88 proof -
```
```    89   interpret ideal I R by fact
```
```    90   show ?thesis
```
```    91     by (intro maximalideal.intro maximalideal_axioms.intro)
```
```    92       (rule is_ideal, rule I_notcarr, rule I_maximal)
```
```    93 qed
```
```    94
```
```    95
```
```    96 subsubsection \<open>Prime Ideals\<close>
```
```    97
```
```    98 locale primeideal = ideal + cring +
```
```    99   assumes I_notcarr: "carrier R \<noteq> I"
```
```   100     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"
```
```   101
```
```   102 lemma (in primeideal) is_primeideal: "primeideal I R"
```
```   103   by (rule primeideal_axioms)
```
```   104
```
```   105 lemma primeidealI:
```
```   106   fixes R (structure)
```
```   107   assumes "ideal I R"
```
```   108     and "cring R"
```
```   109     and I_notcarr: "carrier R \<noteq> I"
```
```   110     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"
```
```   111   shows "primeideal I R"
```
```   112 proof -
```
```   113   interpret ideal I R by fact
```
```   114   interpret cring R by fact
```
```   115   show ?thesis
```
```   116     by (intro primeideal.intro primeideal_axioms.intro)
```
```   117       (rule is_ideal, rule is_cring, rule I_notcarr, rule I_prime)
```
```   118 qed
```
```   119
```
```   120 lemma primeidealI2:
```
```   121   fixes R (structure)
```
```   122   assumes "additive_subgroup I R"
```
```   123     and "cring R"
```
```   124     and I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
```
```   125     and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
```
```   126     and I_notcarr: "carrier R \<noteq> I"
```
```   127     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"
```
```   128   shows "primeideal I R"
```
```   129 proof -
```
```   130   interpret additive_subgroup I R by fact
```
```   131   interpret cring R by fact
```
```   132   show ?thesis apply (intro_locales)
```
```   133     apply (intro ideal_axioms.intro)
```
```   134     apply (erule (1) I_l_closed)
```
```   135     apply (erule (1) I_r_closed)
```
```   136     apply (intro primeideal_axioms.intro)
```
```   137     apply (rule I_notcarr)
```
```   138     apply (erule (2) I_prime)
```
```   139     done
```
```   140 qed
```
```   141
```
```   142
```
```   143 subsection \<open>Special Ideals\<close>
```
```   144
```
```   145 lemma (in ring) zeroideal: "ideal {\<zero>} R"
```
```   146   apply (intro idealI subgroup.intro)
```
```   147         apply (rule is_ring)
```
```   148        apply simp+
```
```   149     apply (fold a_inv_def, simp)
```
```   150    apply simp+
```
```   151   done
```
```   152
```
```   153 lemma (in ring) oneideal: "ideal (carrier R) R"
```
```   154   by (rule idealI) (auto intro: is_ring add.subgroupI)
```
```   155
```
```   156 lemma (in "domain") zeroprimeideal: "primeideal {\<zero>} R"
```
```   157   apply (intro primeidealI)
```
```   158      apply (rule zeroideal)
```
```   159     apply (rule domain.axioms, rule domain_axioms)
```
```   160    defer 1
```
```   161    apply (simp add: integral)
```
```   162 proof (rule ccontr, simp)
```
```   163   assume "carrier R = {\<zero>}"
```
```   164   then have "\<one> = \<zero>" by (rule one_zeroI)
```
```   165   with one_not_zero show False by simp
```
```   166 qed
```
```   167
```
```   168
```
```   169 subsection \<open>General Ideal Properies\<close>
```
```   170
```
```   171 lemma (in ideal) one_imp_carrier:
```
```   172   assumes I_one_closed: "\<one> \<in> I"
```
```   173   shows "I = carrier R"
```
```   174   apply (rule)
```
```   175   apply (rule)
```
```   176   apply (rule a_Hcarr, simp)
```
```   177 proof
```
```   178   fix x
```
```   179   assume xcarr: "x \<in> carrier R"
```
```   180   with I_one_closed have "x \<otimes> \<one> \<in> I" by (intro I_l_closed)
```
```   181   with xcarr show "x \<in> I" by simp
```
```   182 qed
```
```   183
```
```   184 lemma (in ideal) Icarr:
```
```   185   assumes iI: "i \<in> I"
```
```   186   shows "i \<in> carrier R"
```
```   187   using iI by (rule a_Hcarr)
```
```   188
```
```   189
```
```   190 subsection \<open>Intersection of Ideals\<close>
```
```   191
```
```   192 text \<open>\paragraph{Intersection of two ideals} The intersection of any
```
```   193   two ideals is again an ideal in @{term R}\<close>
```
```   194 lemma (in ring) i_intersect:
```
```   195   assumes "ideal I R"
```
```   196   assumes "ideal J R"
```
```   197   shows "ideal (I \<inter> J) R"
```
```   198 proof -
```
```   199   interpret ideal I R by fact
```
```   200   interpret ideal J R by fact
```
```   201   show ?thesis
```
```   202     apply (intro idealI subgroup.intro)
```
```   203           apply (rule is_ring)
```
```   204          apply (force simp add: a_subset)
```
```   205         apply (simp add: a_inv_def[symmetric])
```
```   206        apply simp
```
```   207       apply (simp add: a_inv_def[symmetric])
```
```   208      apply (clarsimp, rule)
```
```   209       apply (fast intro: ideal.I_l_closed ideal.intro assms)+
```
```   210     apply (clarsimp, rule)
```
```   211      apply (fast intro: ideal.I_r_closed ideal.intro assms)+
```
```   212     done
```
```   213 qed
```
```   214
```
```   215 text \<open>The intersection of any Number of Ideals is again
```
```   216         an Ideal in @{term R}\<close>
```
```   217 lemma (in ring) i_Intersect:
```
```   218   assumes Sideals: "\<And>I. I \<in> S \<Longrightarrow> ideal I R"
```
```   219     and notempty: "S \<noteq> {}"
```
```   220   shows "ideal (Inter S) R"
```
```   221   apply (unfold_locales)
```
```   222   apply (simp_all add: Inter_eq)
```
```   223         apply rule unfolding mem_Collect_eq defer 1
```
```   224         apply rule defer 1
```
```   225         apply rule defer 1
```
```   226         apply (fold a_inv_def, rule) defer 1
```
```   227         apply rule defer 1
```
```   228         apply rule defer 1
```
```   229 proof -
```
```   230   fix x y
```
```   231   assume "\<forall>I\<in>S. x \<in> I"
```
```   232   then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   233   assume "\<forall>I\<in>S. y \<in> I"
```
```   234   then have yI: "\<And>I. I \<in> S \<Longrightarrow> y \<in> I" by simp
```
```   235
```
```   236   fix J
```
```   237   assume JS: "J \<in> S"
```
```   238   interpret ideal J R by (rule Sideals[OF JS])
```
```   239   from xI[OF JS] and yI[OF JS] show "x \<oplus> y \<in> J" by (rule a_closed)
```
```   240 next
```
```   241   fix J
```
```   242   assume JS: "J \<in> S"
```
```   243   interpret ideal J R by (rule Sideals[OF JS])
```
```   244   show "\<zero> \<in> J" by simp
```
```   245 next
```
```   246   fix x
```
```   247   assume "\<forall>I\<in>S. x \<in> I"
```
```   248   then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   249
```
```   250   fix J
```
```   251   assume JS: "J \<in> S"
```
```   252   interpret ideal J R by (rule Sideals[OF JS])
```
```   253
```
```   254   from xI[OF JS] show "\<ominus> x \<in> J" by (rule a_inv_closed)
```
```   255 next
```
```   256   fix x y
```
```   257   assume "\<forall>I\<in>S. x \<in> I"
```
```   258   then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   259   assume ycarr: "y \<in> carrier R"
```
```   260
```
```   261   fix J
```
```   262   assume JS: "J \<in> S"
```
```   263   interpret ideal J R by (rule Sideals[OF JS])
```
```   264
```
```   265   from xI[OF JS] and ycarr show "y \<otimes> x \<in> J" by (rule I_l_closed)
```
```   266 next
```
```   267   fix x y
```
```   268   assume "\<forall>I\<in>S. x \<in> I"
```
```   269   then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   270   assume ycarr: "y \<in> carrier R"
```
```   271
```
```   272   fix J
```
```   273   assume JS: "J \<in> S"
```
```   274   interpret ideal J R by (rule Sideals[OF JS])
```
```   275
```
```   276   from xI[OF JS] and ycarr show "x \<otimes> y \<in> J" by (rule I_r_closed)
```
```   277 next
```
```   278   fix x
```
```   279   assume "\<forall>I\<in>S. x \<in> I"
```
```   280   then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
```
```   281
```
```   282   from notempty have "\<exists>I0. I0 \<in> S" by blast
```
```   283   then obtain I0 where I0S: "I0 \<in> S" by auto
```
```   284
```
```   285   interpret ideal I0 R by (rule Sideals[OF I0S])
```
```   286
```
```   287   from xI[OF I0S] have "x \<in> I0" .
```
```   288   with a_subset show "x \<in> carrier R" by fast
```
```   289 next
```
```   290
```
```   291 qed
```
```   292
```
```   293
```
```   294 subsection \<open>Addition of Ideals\<close>
```
```   295
```
```   296 lemma (in ring) add_ideals:
```
```   297   assumes idealI: "ideal I R"
```
```   298       and idealJ: "ideal J R"
```
```   299   shows "ideal (I <+> J) R"
```
```   300   apply (rule ideal.intro)
```
```   301     apply (rule add_additive_subgroups)
```
```   302      apply (intro ideal.axioms[OF idealI])
```
```   303     apply (intro ideal.axioms[OF idealJ])
```
```   304    apply (rule is_ring)
```
```   305   apply (rule ideal_axioms.intro)
```
```   306    apply (simp add: set_add_defs, clarsimp) defer 1
```
```   307    apply (simp add: set_add_defs, clarsimp) defer 1
```
```   308 proof -
```
```   309   fix x i j
```
```   310   assume xcarr: "x \<in> carrier R"
```
```   311     and iI: "i \<in> I"
```
```   312     and jJ: "j \<in> J"
```
```   313   from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
```
```   314   have c: "(i \<oplus> j) \<otimes> x = (i \<otimes> x) \<oplus> (j \<otimes> x)"
```
```   315     by algebra
```
```   316   from xcarr and iI have a: "i \<otimes> x \<in> I"
```
```   317     by (simp add: ideal.I_r_closed[OF idealI])
```
```   318   from xcarr and jJ have b: "j \<otimes> x \<in> J"
```
```   319     by (simp add: ideal.I_r_closed[OF idealJ])
```
```   320   from a b c show "\<exists>ha\<in>I. \<exists>ka\<in>J. (i \<oplus> j) \<otimes> x = ha \<oplus> ka"
```
```   321     by fast
```
```   322 next
```
```   323   fix x i j
```
```   324   assume xcarr: "x \<in> carrier R"
```
```   325     and iI: "i \<in> I"
```
```   326     and jJ: "j \<in> J"
```
```   327   from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
```
```   328   have c: "x \<otimes> (i \<oplus> j) = (x \<otimes> i) \<oplus> (x \<otimes> j)" by algebra
```
```   329   from xcarr and iI have a: "x \<otimes> i \<in> I"
```
```   330     by (simp add: ideal.I_l_closed[OF idealI])
```
```   331   from xcarr and jJ have b: "x \<otimes> j \<in> J"
```
```   332     by (simp add: ideal.I_l_closed[OF idealJ])
```
```   333   from a b c show "\<exists>ha\<in>I. \<exists>ka\<in>J. x \<otimes> (i \<oplus> j) = ha \<oplus> ka"
```
```   334     by fast
```
```   335 qed
```
```   336
```
```   337
```
```   338 subsection (in ring) \<open>Ideals generated by a subset of @{term "carrier R"}\<close>
```
```   339
```
```   340 text \<open>@{term genideal} generates an ideal\<close>
```
```   341 lemma (in ring) genideal_ideal:
```
```   342   assumes Scarr: "S \<subseteq> carrier R"
```
```   343   shows "ideal (Idl S) R"
```
```   344 unfolding genideal_def
```
```   345 proof (rule i_Intersect, fast, simp)
```
```   346   from oneideal and Scarr
```
```   347   show "\<exists>I. ideal I R \<and> S \<le> I" by fast
```
```   348 qed
```
```   349
```
```   350 lemma (in ring) genideal_self:
```
```   351   assumes "S \<subseteq> carrier R"
```
```   352   shows "S \<subseteq> Idl S"
```
```   353   unfolding genideal_def by fast
```
```   354
```
```   355 lemma (in ring) genideal_self':
```
```   356   assumes carr: "i \<in> carrier R"
```
```   357   shows "i \<in> Idl {i}"
```
```   358 proof -
```
```   359   from carr have "{i} \<subseteq> Idl {i}" by (fast intro!: genideal_self)
```
```   360   then show "i \<in> Idl {i}" by fast
```
```   361 qed
```
```   362
```
```   363 text \<open>@{term genideal} generates the minimal ideal\<close>
```
```   364 lemma (in ring) genideal_minimal:
```
```   365   assumes a: "ideal I R"
```
```   366     and b: "S \<subseteq> I"
```
```   367   shows "Idl S \<subseteq> I"
```
```   368   unfolding genideal_def by rule (elim InterD, simp add: a b)
```
```   369
```
```   370 text \<open>Generated ideals and subsets\<close>
```
```   371 lemma (in ring) Idl_subset_ideal:
```
```   372   assumes Iideal: "ideal I R"
```
```   373     and Hcarr: "H \<subseteq> carrier R"
```
```   374   shows "(Idl H \<subseteq> I) = (H \<subseteq> I)"
```
```   375 proof
```
```   376   assume a: "Idl H \<subseteq> I"
```
```   377   from Hcarr have "H \<subseteq> Idl H" by (rule genideal_self)
```
```   378   with a show "H \<subseteq> I" by simp
```
```   379 next
```
```   380   fix x
```
```   381   assume "H \<subseteq> I"
```
```   382   with Iideal have "I \<in> {I. ideal I R \<and> H \<subseteq> I}" by fast
```
```   383   then show "Idl H \<subseteq> I" unfolding genideal_def by fast
```
```   384 qed
```
```   385
```
```   386 lemma (in ring) subset_Idl_subset:
```
```   387   assumes Icarr: "I \<subseteq> carrier R"
```
```   388     and HI: "H \<subseteq> I"
```
```   389   shows "Idl H \<subseteq> Idl I"
```
```   390 proof -
```
```   391   from HI and genideal_self[OF Icarr] have HIdlI: "H \<subseteq> Idl I"
```
```   392     by fast
```
```   393
```
```   394   from Icarr have Iideal: "ideal (Idl I) R"
```
```   395     by (rule genideal_ideal)
```
```   396   from HI and Icarr have "H \<subseteq> carrier R"
```
```   397     by fast
```
```   398   with Iideal have "(H \<subseteq> Idl I) = (Idl H \<subseteq> Idl I)"
```
```   399     by (rule Idl_subset_ideal[symmetric])
```
```   400
```
```   401   with HIdlI show "Idl H \<subseteq> Idl I" by simp
```
```   402 qed
```
```   403
```
```   404 lemma (in ring) Idl_subset_ideal':
```
```   405   assumes acarr: "a \<in> carrier R" and bcarr: "b \<in> carrier R"
```
```   406   shows "(Idl {a} \<subseteq> Idl {b}) = (a \<in> Idl {b})"
```
```   407   apply (subst Idl_subset_ideal[OF genideal_ideal[of "{b}"], of "{a}"])
```
```   408     apply (fast intro: bcarr, fast intro: acarr)
```
```   409   apply fast
```
```   410   done
```
```   411
```
```   412 lemma (in ring) genideal_zero: "Idl {\<zero>} = {\<zero>}"
```
```   413   apply rule
```
```   414    apply (rule genideal_minimal[OF zeroideal], simp)
```
```   415   apply (simp add: genideal_self')
```
```   416   done
```
```   417
```
```   418 lemma (in ring) genideal_one: "Idl {\<one>} = carrier R"
```
```   419 proof -
```
```   420   interpret ideal "Idl {\<one>}" "R" by (rule genideal_ideal) fast
```
```   421   show "Idl {\<one>} = carrier R"
```
```   422   apply (rule, rule a_subset)
```
```   423   apply (simp add: one_imp_carrier genideal_self')
```
```   424   done
```
```   425 qed
```
```   426
```
```   427
```
```   428 text \<open>Generation of Principal Ideals in Commutative Rings\<close>
```
```   429
```
```   430 definition cgenideal :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set"  ("PIdl\<index> _"  79)
```
```   431   where "cgenideal R a = {x \<otimes>\<^bsub>R\<^esub> a | x. x \<in> carrier R}"
```
```   432
```
```   433 text \<open>genhideal (?) really generates an ideal\<close>
```
```   434 lemma (in cring) cgenideal_ideal:
```
```   435   assumes acarr: "a \<in> carrier R"
```
```   436   shows "ideal (PIdl a) R"
```
```   437   apply (unfold cgenideal_def)
```
```   438   apply (rule idealI[OF is_ring])
```
```   439      apply (rule subgroup.intro)
```
```   440         apply simp_all
```
```   441         apply (blast intro: acarr)
```
```   442         apply clarsimp defer 1
```
```   443         defer 1
```
```   444         apply (fold a_inv_def, clarsimp) defer 1
```
```   445         apply clarsimp defer 1
```
```   446         apply clarsimp defer 1
```
```   447 proof -
```
```   448   fix x y
```
```   449   assume xcarr: "x \<in> carrier R"
```
```   450     and ycarr: "y \<in> carrier R"
```
```   451   note carr = acarr xcarr ycarr
```
```   452
```
```   453   from carr have "x \<otimes> a \<oplus> y \<otimes> a = (x \<oplus> y) \<otimes> a"
```
```   454     by (simp add: l_distr)
```
```   455   with carr show "\<exists>z. x \<otimes> a \<oplus> y \<otimes> a = z \<otimes> a \<and> z \<in> carrier R"
```
```   456     by fast
```
```   457 next
```
```   458   from l_null[OF acarr, symmetric] and zero_closed
```
```   459   show "\<exists>x. \<zero> = x \<otimes> a \<and> x \<in> carrier R" by fast
```
```   460 next
```
```   461   fix x
```
```   462   assume xcarr: "x \<in> carrier R"
```
```   463   note carr = acarr xcarr
```
```   464
```
```   465   from carr have "\<ominus> (x \<otimes> a) = (\<ominus> x) \<otimes> a"
```
```   466     by (simp add: l_minus)
```
```   467   with carr show "\<exists>z. \<ominus> (x \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R"
```
```   468     by fast
```
```   469 next
```
```   470   fix x y
```
```   471   assume xcarr: "x \<in> carrier R"
```
```   472      and ycarr: "y \<in> carrier R"
```
```   473   note carr = acarr xcarr ycarr
```
```   474
```
```   475   from carr have "y \<otimes> a \<otimes> x = (y \<otimes> x) \<otimes> a"
```
```   476     by (simp add: m_assoc) (simp add: m_comm)
```
```   477   with carr show "\<exists>z. y \<otimes> a \<otimes> x = z \<otimes> a \<and> z \<in> carrier R"
```
```   478     by fast
```
```   479 next
```
```   480   fix x y
```
```   481   assume xcarr: "x \<in> carrier R"
```
```   482      and ycarr: "y \<in> carrier R"
```
```   483   note carr = acarr xcarr ycarr
```
```   484
```
```   485   from carr have "x \<otimes> (y \<otimes> a) = (x \<otimes> y) \<otimes> a"
```
```   486     by (simp add: m_assoc)
```
```   487   with carr show "\<exists>z. x \<otimes> (y \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R"
```
```   488     by fast
```
```   489 qed
```
```   490
```
```   491 lemma (in ring) cgenideal_self:
```
```   492   assumes icarr: "i \<in> carrier R"
```
```   493   shows "i \<in> PIdl i"
```
```   494   unfolding cgenideal_def
```
```   495 proof simp
```
```   496   from icarr have "i = \<one> \<otimes> i"
```
```   497     by simp
```
```   498   with icarr show "\<exists>x. i = x \<otimes> i \<and> x \<in> carrier R"
```
```   499     by fast
```
```   500 qed
```
```   501
```
```   502 text \<open>@{const "cgenideal"} is minimal\<close>
```
```   503
```
```   504 lemma (in ring) cgenideal_minimal:
```
```   505   assumes "ideal J R"
```
```   506   assumes aJ: "a \<in> J"
```
```   507   shows "PIdl a \<subseteq> J"
```
```   508 proof -
```
```   509   interpret ideal J R by fact
```
```   510   show ?thesis
```
```   511     unfolding cgenideal_def
```
```   512     apply rule
```
```   513     apply clarify
```
```   514     using aJ
```
```   515     apply (erule I_l_closed)
```
```   516     done
```
```   517 qed
```
```   518
```
```   519 lemma (in cring) cgenideal_eq_genideal:
```
```   520   assumes icarr: "i \<in> carrier R"
```
```   521   shows "PIdl i = Idl {i}"
```
```   522   apply rule
```
```   523    apply (intro cgenideal_minimal)
```
```   524     apply (rule genideal_ideal, fast intro: icarr)
```
```   525    apply (rule genideal_self', fast intro: icarr)
```
```   526   apply (intro genideal_minimal)
```
```   527    apply (rule cgenideal_ideal [OF icarr])
```
```   528   apply (simp, rule cgenideal_self [OF icarr])
```
```   529   done
```
```   530
```
```   531 lemma (in cring) cgenideal_eq_rcos: "PIdl i = carrier R #> i"
```
```   532   unfolding cgenideal_def r_coset_def by fast
```
```   533
```
```   534 lemma (in cring) cgenideal_is_principalideal:
```
```   535   assumes icarr: "i \<in> carrier R"
```
```   536   shows "principalideal (PIdl i) R"
```
```   537   apply (rule principalidealI)
```
```   538   apply (rule cgenideal_ideal [OF icarr])
```
```   539 proof -
```
```   540   from icarr have "PIdl i = Idl {i}"
```
```   541     by (rule cgenideal_eq_genideal)
```
```   542   with icarr show "\<exists>i'\<in>carrier R. PIdl i = Idl {i'}"
```
```   543     by fast
```
```   544 qed
```
```   545
```
```   546
```
```   547 subsection \<open>Union of Ideals\<close>
```
```   548
```
```   549 lemma (in ring) union_genideal:
```
```   550   assumes idealI: "ideal I R"
```
```   551     and idealJ: "ideal J R"
```
```   552   shows "Idl (I \<union> J) = I <+> J"
```
```   553   apply rule
```
```   554    apply (rule ring.genideal_minimal)
```
```   555      apply (rule is_ring)
```
```   556     apply (rule add_ideals[OF idealI idealJ])
```
```   557    apply (rule)
```
```   558    apply (simp add: set_add_defs) apply (elim disjE) defer 1 defer 1
```
```   559    apply (rule) apply (simp add: set_add_defs genideal_def) apply clarsimp defer 1
```
```   560 proof -
```
```   561   fix x
```
```   562   assume xI: "x \<in> I"
```
```   563   have ZJ: "\<zero> \<in> J"
```
```   564     by (intro additive_subgroup.zero_closed) (rule ideal.axioms[OF idealJ])
```
```   565   from ideal.Icarr[OF idealI xI] have "x = x \<oplus> \<zero>"
```
```   566     by algebra
```
```   567   with xI and ZJ show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k"
```
```   568     by fast
```
```   569 next
```
```   570   fix x
```
```   571   assume xJ: "x \<in> J"
```
```   572   have ZI: "\<zero> \<in> I"
```
```   573     by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealI])
```
```   574   from ideal.Icarr[OF idealJ xJ] have "x = \<zero> \<oplus> x"
```
```   575     by algebra
```
```   576   with ZI and xJ show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k"
```
```   577     by fast
```
```   578 next
```
```   579   fix i j K
```
```   580   assume iI: "i \<in> I"
```
```   581     and jJ: "j \<in> J"
```
```   582     and idealK: "ideal K R"
```
```   583     and IK: "I \<subseteq> K"
```
```   584     and JK: "J \<subseteq> K"
```
```   585   from iI and IK have iK: "i \<in> K" by fast
```
```   586   from jJ and JK have jK: "j \<in> K" by fast
```
```   587   from iK and jK show "i \<oplus> j \<in> K"
```
```   588     by (intro additive_subgroup.a_closed) (rule ideal.axioms[OF idealK])
```
```   589 qed
```
```   590
```
```   591
```
```   592 subsection \<open>Properties of Principal Ideals\<close>
```
```   593
```
```   594 text \<open>@{text "\<zero>"} generates the zero ideal\<close>
```
```   595 lemma (in ring) zero_genideal: "Idl {\<zero>} = {\<zero>}"
```
```   596   apply rule
```
```   597   apply (simp add: genideal_minimal zeroideal)
```
```   598   apply (fast intro!: genideal_self)
```
```   599   done
```
```   600
```
```   601 text \<open>@{text "\<one>"} generates the unit ideal\<close>
```
```   602 lemma (in ring) one_genideal: "Idl {\<one>} = carrier R"
```
```   603 proof -
```
```   604   have "\<one> \<in> Idl {\<one>}"
```
```   605     by (simp add: genideal_self')
```
```   606   then show "Idl {\<one>} = carrier R"
```
```   607     by (intro ideal.one_imp_carrier) (fast intro: genideal_ideal)
```
```   608 qed
```
```   609
```
```   610
```
```   611 text \<open>The zero ideal is a principal ideal\<close>
```
```   612 corollary (in ring) zeropideal: "principalideal {\<zero>} R"
```
```   613   apply (rule principalidealI)
```
```   614    apply (rule zeroideal)
```
```   615   apply (blast intro!: zero_genideal[symmetric])
```
```   616   done
```
```   617
```
```   618 text \<open>The unit ideal is a principal ideal\<close>
```
```   619 corollary (in ring) onepideal: "principalideal (carrier R) R"
```
```   620   apply (rule principalidealI)
```
```   621    apply (rule oneideal)
```
```   622   apply (blast intro!: one_genideal[symmetric])
```
```   623   done
```
```   624
```
```   625
```
```   626 text \<open>Every principal ideal is a right coset of the carrier\<close>
```
```   627 lemma (in principalideal) rcos_generate:
```
```   628   assumes "cring R"
```
```   629   shows "\<exists>x\<in>I. I = carrier R #> x"
```
```   630 proof -
```
```   631   interpret cring R by fact
```
```   632   from generate obtain i where icarr: "i \<in> carrier R" and I1: "I = Idl {i}"
```
```   633     by fast+
```
```   634
```
```   635   from icarr and genideal_self[of "{i}"] have "i \<in> Idl {i}"
```
```   636     by fast
```
```   637   then have iI: "i \<in> I" by (simp add: I1)
```
```   638
```
```   639   from I1 icarr have I2: "I = PIdl i"
```
```   640     by (simp add: cgenideal_eq_genideal)
```
```   641
```
```   642   have "PIdl i = carrier R #> i"
```
```   643     unfolding cgenideal_def r_coset_def by fast
```
```   644
```
```   645   with I2 have "I = carrier R #> i"
```
```   646     by simp
```
```   647
```
```   648   with iI show "\<exists>x\<in>I. I = carrier R #> x"
```
```   649     by fast
```
```   650 qed
```
```   651
```
```   652
```
```   653 subsection \<open>Prime Ideals\<close>
```
```   654
```
```   655 lemma (in ideal) primeidealCD:
```
```   656   assumes "cring R"
```
```   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   interpret cring R by fact
```
```   661   assume InR: "carrier R \<noteq> I"
```
```   662     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)"
```
```   663   then have 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"
```
```   664     by simp
```
```   665   have "primeideal I R"
```
```   666     apply (rule primeideal.intro [OF is_ideal is_cring])
```
```   667     apply (rule primeideal_axioms.intro)
```
```   668      apply (rule InR)
```
```   669     apply (erule (2) I_prime)
```
```   670     done
```
```   671   with notprime show False by simp
```
```   672 qed
```
```   673
```
```   674 lemma (in ideal) primeidealCE:
```
```   675   assumes "cring R"
```
```   676   assumes notprime: "\<not> primeideal I R"
```
```   677   obtains "carrier R = I"
```
```   678     | "\<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 proof -
```
```   680   interpret R: cring R by fact
```
```   681   assume "carrier R = I ==> thesis"
```
```   682     and "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I \<Longrightarrow> thesis"
```
```   683   then show thesis using primeidealCD [OF R.is_cring notprime] by blast
```
```   684 qed
```
```   685
```
```   686 text \<open>If @{text "{\<zero>}"} is a prime ideal of a commutative ring, the ring is a domain\<close>
```
```   687 lemma (in cring) zeroprimeideal_domainI:
```
```   688   assumes pi: "primeideal {\<zero>} R"
```
```   689   shows "domain R"
```
```   690   apply (rule domain.intro, rule is_cring)
```
```   691   apply (rule domain_axioms.intro)
```
```   692 proof (rule ccontr, simp)
```
```   693   interpret primeideal "{\<zero>}" "R" by (rule pi)
```
```   694   assume "\<one> = \<zero>"
```
```   695   then have "carrier R = {\<zero>}" by (rule one_zeroD)
```
```   696   from this[symmetric] and I_notcarr show False
```
```   697     by simp
```
```   698 next
```
```   699   interpret primeideal "{\<zero>}" "R" by (rule pi)
```
```   700   fix a b
```
```   701   assume ab: "a \<otimes> b = \<zero>" and carr: "a \<in> carrier R" "b \<in> carrier R"
```
```   702   from ab have abI: "a \<otimes> b \<in> {\<zero>}"
```
```   703     by fast
```
```   704   with carr have "a \<in> {\<zero>} \<or> b \<in> {\<zero>}"
```
```   705     by (rule I_prime)
```
```   706   then show "a = \<zero> \<or> b = \<zero>" by simp
```
```   707 qed
```
```   708
```
```   709 corollary (in cring) domain_eq_zeroprimeideal: "domain R = primeideal {\<zero>} R"
```
```   710   apply rule
```
```   711    apply (erule domain.zeroprimeideal)
```
```   712   apply (erule zeroprimeideal_domainI)
```
```   713   done
```
```   714
```
```   715
```
```   716 subsection \<open>Maximal Ideals\<close>
```
```   717
```
```   718 lemma (in ideal) helper_I_closed:
```
```   719   assumes carr: "a \<in> carrier R" "x \<in> carrier R" "y \<in> carrier R"
```
```   720     and axI: "a \<otimes> x \<in> I"
```
```   721   shows "a \<otimes> (x \<otimes> y) \<in> I"
```
```   722 proof -
```
```   723   from axI and carr have "(a \<otimes> x) \<otimes> y \<in> I"
```
```   724     by (simp add: I_r_closed)
```
```   725   also from carr have "(a \<otimes> x) \<otimes> y = a \<otimes> (x \<otimes> y)"
```
```   726     by (simp add: m_assoc)
```
```   727   finally show "a \<otimes> (x \<otimes> y) \<in> I" .
```
```   728 qed
```
```   729
```
```   730 lemma (in ideal) helper_max_prime:
```
```   731   assumes "cring R"
```
```   732   assumes acarr: "a \<in> carrier R"
```
```   733   shows "ideal {x\<in>carrier R. a \<otimes> x \<in> I} R"
```
```   734 proof -
```
```   735   interpret cring R by fact
```
```   736   show ?thesis 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 show "a \<otimes> (\<ominus>x) \<in> I" .
```
```   754     from acarr have "a \<otimes> \<zero> = \<zero>" by simp
```
```   755   next
```
```   756     fix x y
```
```   757     assume xcarr: "x \<in> carrier R"
```
```   758       and ycarr: "y \<in> carrier R"
```
```   759       and ayI: "a \<otimes> y \<in> I"
```
```   760     from ayI and acarr xcarr ycarr have "a \<otimes> (y \<otimes> x) \<in> I"
```
```   761       by (simp add: helper_I_closed)
```
```   762     moreover
```
```   763     from xcarr ycarr have "y \<otimes> x = x \<otimes> y"
```
```   764       by (simp add: m_comm)
```
```   765     ultimately
```
```   766     show "a \<otimes> (x \<otimes> y) \<in> I" by simp
```
```   767   qed
```
```   768 qed
```
```   769
```
```   770 text \<open>In a cring every maximal ideal is prime\<close>
```
```   771 lemma (in cring) maximalideal_is_prime:
```
```   772   assumes "maximalideal I R"
```
```   773   shows "primeideal I R"
```
```   774 proof -
```
```   775   interpret maximalideal I R by fact
```
```   776   show ?thesis apply (rule ccontr)
```
```   777     apply (rule primeidealCE)
```
```   778     apply (rule is_cring)
```
```   779     apply assumption
```
```   780     apply (simp add: I_notcarr)
```
```   781   proof -
```
```   782     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"
```
```   783     then obtain a b where
```
```   784       acarr: "a \<in> carrier R" and
```
```   785       bcarr: "b \<in> carrier R" and
```
```   786       abI: "a \<otimes> b \<in> I" and
```
```   787       anI: "a \<notin> I" and
```
```   788       bnI: "b \<notin> I" by fast
```
```   789     def J \<equiv> "{x\<in>carrier R. a \<otimes> x \<in> I}"
```
```   790
```
```   791     from is_cring and acarr have idealJ: "ideal J R"
```
```   792       unfolding J_def by (rule helper_max_prime)
```
```   793
```
```   794     have IsubJ: "I \<subseteq> J"
```
```   795     proof
```
```   796       fix x
```
```   797       assume xI: "x \<in> I"
```
```   798       with acarr have "a \<otimes> x \<in> I"
```
```   799         by (intro I_l_closed)
```
```   800       with xI[THEN a_Hcarr] show "x \<in> J"
```
```   801         unfolding J_def by fast
```
```   802     qed
```
```   803
```
```   804     from abI and acarr bcarr have "b \<in> J"
```
```   805       unfolding J_def by fast
```
```   806     with bnI have JnI: "J \<noteq> I" by fast
```
```   807     from acarr
```
```   808     have "a = a \<otimes> \<one>" by algebra
```
```   809     with anI have "a \<otimes> \<one> \<notin> I" by simp
```
```   810     with one_closed have "\<one> \<notin> J"
```
```   811       unfolding J_def by fast
```
```   812     then have Jncarr: "J \<noteq> carrier R" by fast
```
```   813
```
```   814     interpret ideal J R by (rule idealJ)
```
```   815
```
```   816     have "J = I \<or> J = carrier R"
```
```   817       apply (intro I_maximal)
```
```   818       apply (rule idealJ)
```
```   819       apply (rule IsubJ)
```
```   820       apply (rule a_subset)
```
```   821       done
```
```   822
```
```   823     with JnI and Jncarr show False by simp
```
```   824   qed
```
```   825 qed
```
```   826
```
```   827
```
```   828 subsection \<open>Derived Theorems\<close>
```
```   829
```
```   830 --"A non-zero cring that has only the two trivial ideals is a field"
```
```   831 lemma (in cring) trivialideals_fieldI:
```
```   832   assumes carrnzero: "carrier R \<noteq> {\<zero>}"
```
```   833     and haveideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
```
```   834   shows "field R"
```
```   835   apply (rule cring_fieldI)
```
```   836   apply (rule, rule, rule)
```
```   837    apply (erule Units_closed)
```
```   838   defer 1
```
```   839     apply rule
```
```   840   defer 1
```
```   841 proof (rule ccontr, simp)
```
```   842   assume zUnit: "\<zero> \<in> Units R"
```
```   843   then have a: "\<zero> \<otimes> inv \<zero> = \<one>" by (rule Units_r_inv)
```
```   844   from zUnit have "\<zero> \<otimes> inv \<zero> = \<zero>"
```
```   845     by (intro l_null) (rule Units_inv_closed)
```
```   846   with a[symmetric] have "\<one> = \<zero>" by simp
```
```   847   then have "carrier R = {\<zero>}" by (rule one_zeroD)
```
```   848   with carrnzero show False by simp
```
```   849 next
```
```   850   fix x
```
```   851   assume xcarr': "x \<in> carrier R - {\<zero>}"
```
```   852   then have xcarr: "x \<in> carrier R" by fast
```
```   853   from xcarr' have xnZ: "x \<noteq> \<zero>" by fast
```
```   854   from xcarr have xIdl: "ideal (PIdl x) R"
```
```   855     by (intro cgenideal_ideal) fast
```
```   856
```
```   857   from xcarr have "x \<in> PIdl x"
```
```   858     by (intro cgenideal_self) fast
```
```   859   with xnZ have "PIdl x \<noteq> {\<zero>}" by fast
```
```   860   with haveideals have "PIdl x = carrier R"
```
```   861     by (blast intro!: xIdl)
```
```   862   then have "\<one> \<in> PIdl x" by simp
```
```   863   then have "\<exists>y. \<one> = y \<otimes> x \<and> y \<in> carrier R"
```
```   864     unfolding cgenideal_def by blast
```
```   865   then obtain y where ycarr: " y \<in> carrier R" and ylinv: "\<one> = y \<otimes> x"
```
```   866     by fast+
```
```   867   from ylinv and xcarr ycarr have yrinv: "\<one> = x \<otimes> y"
```
```   868     by (simp add: m_comm)
```
```   869   from ycarr and ylinv[symmetric] and yrinv[symmetric]
```
```   870   have "\<exists>y \<in> carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
```
```   871   with xcarr show "x \<in> Units R"
```
```   872     unfolding Units_def by fast
```
```   873 qed
```
```   874
```
```   875 lemma (in field) all_ideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
```
```   876   apply (rule, rule)
```
```   877 proof -
```
```   878   fix I
```
```   879   assume a: "I \<in> {I. ideal I R}"
```
```   880   then interpret ideal I R by simp
```
```   881
```
```   882   show "I \<in> {{\<zero>}, carrier R}"
```
```   883   proof (cases "\<exists>a. a \<in> I - {\<zero>}")
```
```   884     case True
```
```   885     then obtain a where aI: "a \<in> I" and anZ: "a \<noteq> \<zero>"
```
```   886       by fast+
```
```   887     from aI[THEN a_Hcarr] anZ have aUnit: "a \<in> Units R"
```
```   888       by (simp add: field_Units)
```
```   889     then have a: "a \<otimes> inv a = \<one>" by (rule Units_r_inv)
```
```   890     from aI and aUnit have "a \<otimes> inv a \<in> I"
```
```   891       by (simp add: I_r_closed del: Units_r_inv)
```
```   892     then have oneI: "\<one> \<in> I" by (simp add: a[symmetric])
```
```   893
```
```   894     have "carrier R \<subseteq> I"
```
```   895     proof
```
```   896       fix x
```
```   897       assume xcarr: "x \<in> carrier R"
```
```   898       with oneI have "\<one> \<otimes> x \<in> I" by (rule I_r_closed)
```
```   899       with xcarr show "x \<in> I" by simp
```
```   900     qed
```
```   901     with a_subset have "I = carrier R" by fast
```
```   902     then show "I \<in> {{\<zero>}, carrier R}" by fast
```
```   903   next
```
```   904     case False
```
```   905     then have IZ: "\<And>a. a \<in> I \<Longrightarrow> a = \<zero>" by simp
```
```   906
```
```   907     have a: "I \<subseteq> {\<zero>}"
```
```   908     proof
```
```   909       fix x
```
```   910       assume "x \<in> I"
```
```   911       then have "x = \<zero>" by (rule IZ)
```
```   912       then show "x \<in> {\<zero>}" by fast
```
```   913     qed
```
```   914
```
```   915     have "\<zero> \<in> I" by simp
```
```   916     then have "{\<zero>} \<subseteq> I" by fast
```
```   917
```
```   918     with a have "I = {\<zero>}" by fast
```
```   919     then show "I \<in> {{\<zero>}, carrier R}" by fast
```
```   920   qed
```
```   921 qed (simp add: zeroideal oneideal)
```
```   922
```
```   923 --"Jacobson Theorem 2.2"
```
```   924 lemma (in cring) trivialideals_eq_field:
```
```   925   assumes carrnzero: "carrier R \<noteq> {\<zero>}"
```
```   926   shows "({I. ideal I R} = {{\<zero>}, carrier R}) = field R"
```
```   927   by (fast intro!: trivialideals_fieldI[OF carrnzero] field.all_ideals)
```
```   928
```
```   929
```
```   930 text \<open>Like zeroprimeideal for domains\<close>
```
```   931 lemma (in field) zeromaximalideal: "maximalideal {\<zero>} R"
```
```   932   apply (rule maximalidealI)
```
```   933     apply (rule zeroideal)
```
```   934 proof-
```
```   935   from one_not_zero have "\<one> \<notin> {\<zero>}" by simp
```
```   936   with one_closed show "carrier R \<noteq> {\<zero>}" by fast
```
```   937 next
```
```   938   fix J
```
```   939   assume Jideal: "ideal J R"
```
```   940   then have "J \<in> {I. ideal I R}" by fast
```
```   941   with all_ideals show "J = {\<zero>} \<or> J = carrier R"
```
```   942     by simp
```
```   943 qed
```
```   944
```
```   945 lemma (in cring) zeromaximalideal_fieldI:
```
```   946   assumes zeromax: "maximalideal {\<zero>} R"
```
```   947   shows "field R"
```
```   948   apply (rule trivialideals_fieldI, rule maximalideal.I_notcarr[OF zeromax])
```
```   949   apply rule apply clarsimp defer 1
```
```   950    apply (simp add: zeroideal oneideal)
```
```   951 proof -
```
```   952   fix J
```
```   953   assume Jn0: "J \<noteq> {\<zero>}"
```
```   954     and idealJ: "ideal J R"
```
```   955   interpret ideal J R by (rule idealJ)
```
```   956   have "{\<zero>} \<subseteq> J" by (rule ccontr) simp
```
```   957   from zeromax and idealJ and this and a_subset
```
```   958   have "J = {\<zero>} \<or> J = carrier R"
```
```   959     by (rule maximalideal.I_maximal)
```
```   960   with Jn0 show "J = carrier R"
```
```   961     by simp
```
```   962 qed
```
```   963
```
```   964 lemma (in cring) zeromaximalideal_eq_field: "maximalideal {\<zero>} R = field R"
```
```   965   apply rule
```
```   966    apply (erule zeromaximalideal_fieldI)
```
```   967   apply (erule field.zeromaximalideal)
```
```   968   done
```
```   969
```
```   970 end
```