src/HOL/Algebra/QuotRing.thy
author wenzelm
Sun Mar 21 15:57:40 2010 +0100 (2010-03-21)
changeset 35847 19f1f7066917
parent 29242 e190bc2a5399
child 35848 5443079512ea
permissions -rw-r--r--
eliminated old constdefs;
     1 (*
     2   Title:     HOL/Algebra/QuotRing.thy
     3   Author:    Stephan Hohe
     4 *)
     5 
     6 theory QuotRing
     7 imports RingHom
     8 begin
     9 
    10 section {* Quotient Rings *}
    11 
    12 subsection {* Multiplication on Cosets *}
    13 
    14 definition
    15   rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
    16     ("[mod _:] _ \<Otimes>\<index> _" [81,81,81] 80)
    17   where "rcoset_mult R I A B \<equiv> \<Union>a\<in>A. \<Union>b\<in>B. I +>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b)"
    18 
    19 
    20 text {* @{const "rcoset_mult"} fulfils the properties required by
    21   congruences *}
    22 lemma (in ideal) rcoset_mult_add:
    23   "\<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
    24 apply rule
    25 apply (rule, simp add: rcoset_mult_def, clarsimp)
    26 defer 1
    27 apply (rule, simp add: rcoset_mult_def)
    28 defer 1
    29 proof -
    30   fix z x' y'
    31   assume carr: "x \<in> carrier R" "y \<in> carrier R"
    32      and x'rcos: "x' \<in> I +> x"
    33      and y'rcos: "y' \<in> I +> y"
    34      and zrcos: "z \<in> I +> x' \<otimes> y'"
    35 
    36   from x'rcos 
    37       have "\<exists>h\<in>I. x' = h \<oplus> x" by (simp add: a_r_coset_def r_coset_def)
    38   from this obtain hx
    39       where hxI: "hx \<in> I"
    40       and x': "x' = hx \<oplus> x"
    41       by fast+
    42   
    43   from y'rcos
    44       have "\<exists>h\<in>I. y' = h \<oplus> y" by (simp add: a_r_coset_def r_coset_def)
    45   from this
    46       obtain hy
    47       where hyI: "hy \<in> I"
    48       and y': "y' = hy \<oplus> y"
    49       by fast+
    50 
    51   from zrcos
    52       have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')" by (simp add: a_r_coset_def r_coset_def)
    53   from this
    54       obtain hz
    55       where hzI: "hz \<in> I"
    56       and z: "z = hz \<oplus> (x' \<otimes> y')"
    57       by fast+
    58 
    59   note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
    60 
    61   from z have "z = hz \<oplus> (x' \<otimes> y')" .
    62   also from x' y'
    63       have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
    64   also from carr
    65       have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
    66   finally
    67       have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
    68 
    69   from hxI hyI hzI carr
    70       have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"  by (simp add: I_l_closed I_r_closed)
    71 
    72   from this and z2
    73       have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
    74   thus "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
    75 next
    76   fix z
    77   assume xcarr: "x \<in> carrier R"
    78      and ycarr: "y \<in> carrier R"
    79      and zrcos: "z \<in> I +> x \<otimes> y"
    80   from xcarr
    81       have xself: "x \<in> I +> x" by (intro a_rcos_self)
    82   from ycarr
    83       have yself: "y \<in> I +> y" by (intro a_rcos_self)
    84 
    85   from xself and yself and zrcos
    86       show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" by fast
    87 qed
    88 
    89 
    90 subsection {* Quotient Ring Definition *}
    91 
    92 definition
    93   FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"  (infixl "Quot" 65)
    94   where "FactRing R I \<equiv>
    95     \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), zero = I, add = set_add R\<rparr>"
    96 
    97 
    98 subsection {* Factorization over General Ideals *}
    99 
   100 text {* The quotient is a ring *}
   101 lemma (in ideal) quotient_is_ring:
   102   shows "ring (R Quot I)"
   103 apply (rule ringI)
   104    --{* abelian group *}
   105    apply (rule comm_group_abelian_groupI)
   106    apply (simp add: FactRing_def)
   107    apply (rule a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
   108   --{* mult monoid *}
   109   apply (rule monoidI)
   110       apply (simp_all add: FactRing_def A_RCOSETS_def RCOSETS_def
   111              a_r_coset_def[symmetric])
   112       --{* mult closed *}
   113       apply (clarify)
   114       apply (simp add: rcoset_mult_add, fast)
   115      --{* mult @{text one_closed} *}
   116      apply (force intro: one_closed)
   117     --{* mult assoc *}
   118     apply clarify
   119     apply (simp add: rcoset_mult_add m_assoc)
   120    --{* mult one *}
   121    apply clarify
   122    apply (simp add: rcoset_mult_add l_one)
   123   apply clarify
   124   apply (simp add: rcoset_mult_add r_one)
   125  --{* distr *}
   126  apply clarify
   127  apply (simp add: rcoset_mult_add a_rcos_sum l_distr)
   128 apply clarify
   129 apply (simp add: rcoset_mult_add a_rcos_sum r_distr)
   130 done
   131 
   132 
   133 text {* This is a ring homomorphism *}
   134 
   135 lemma (in ideal) rcos_ring_hom:
   136   "(op +> I) \<in> ring_hom R (R Quot I)"
   137 apply (rule ring_hom_memI)
   138    apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
   139   apply (simp add: FactRing_def rcoset_mult_add)
   140  apply (simp add: FactRing_def a_rcos_sum)
   141 apply (simp add: FactRing_def)
   142 done
   143 
   144 lemma (in ideal) rcos_ring_hom_ring:
   145   "ring_hom_ring R (R Quot I) (op +> I)"
   146 apply (rule ring_hom_ringI)
   147      apply (rule is_ring, rule quotient_is_ring)
   148    apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
   149   apply (simp add: FactRing_def rcoset_mult_add)
   150  apply (simp add: FactRing_def a_rcos_sum)
   151 apply (simp add: FactRing_def)
   152 done
   153 
   154 text {* The quotient of a cring is also commutative *}
   155 lemma (in ideal) quotient_is_cring:
   156   assumes "cring R"
   157   shows "cring (R Quot I)"
   158 proof -
   159   interpret cring R by fact
   160   show ?thesis apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
   161   apply (rule quotient_is_ring)
   162  apply (rule ring.axioms[OF quotient_is_ring])
   163 apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
   164 apply clarify
   165 apply (simp add: rcoset_mult_add m_comm)
   166 done
   167 qed
   168 
   169 text {* Cosets as a ring homomorphism on crings *}
   170 lemma (in ideal) rcos_ring_hom_cring:
   171   assumes "cring R"
   172   shows "ring_hom_cring R (R Quot I) (op +> I)"
   173 proof -
   174   interpret cring R by fact
   175   show ?thesis apply (rule ring_hom_cringI)
   176   apply (rule rcos_ring_hom_ring)
   177  apply (rule is_cring)
   178 apply (rule quotient_is_cring)
   179 apply (rule is_cring)
   180 done
   181 qed
   182 
   183 subsection {* Factorization over Prime Ideals *}
   184 
   185 text {* The quotient ring generated by a prime ideal is a domain *}
   186 lemma (in primeideal) quotient_is_domain:
   187   shows "domain (R Quot I)"
   188 apply (rule domain.intro)
   189  apply (rule quotient_is_cring, rule is_cring)
   190 apply (rule domain_axioms.intro)
   191  apply (simp add: FactRing_def) defer 1
   192  apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
   193  apply (simp add: rcoset_mult_add) defer 1
   194 proof (rule ccontr, clarsimp)
   195   assume "I +> \<one> = I"
   196   hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
   197   hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
   198   from this and a_subset
   199       have "I = carrier R" by fast
   200   from this and I_notcarr
   201       show "False" by fast
   202 next
   203   fix x y
   204   assume carr: "x \<in> carrier R" "y \<in> carrier R"
   205      and a: "I +> x \<otimes> y = I"
   206      and b: "I +> y \<noteq> I"
   207 
   208   have ynI: "y \<notin> I"
   209   proof (rule ccontr, simp)
   210     assume "y \<in> I"
   211     hence "I +> y = I" by (rule a_rcos_const)
   212     from this and b
   213         show "False" by simp
   214   qed
   215 
   216   from carr
   217       have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
   218   from this
   219       have xyI: "x \<otimes> y \<in> I" by (simp add: a)
   220 
   221   from xyI and carr
   222       have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
   223   from this and ynI
   224       have "x \<in> I" by fast
   225   thus "I +> x = I" by (rule a_rcos_const)
   226 qed
   227 
   228 text {* Generating right cosets of a prime ideal is a homomorphism
   229         on commutative rings *}
   230 lemma (in primeideal) rcos_ring_hom_cring:
   231   shows "ring_hom_cring R (R Quot I) (op +> I)"
   232 by (rule rcos_ring_hom_cring, rule is_cring)
   233 
   234 
   235 subsection {* Factorization over Maximal Ideals *}
   236 
   237 text {* In a commutative ring, the quotient ring over a maximal ideal
   238         is a field.
   239         The proof follows ``W. Adkins, S. Weintraub: Algebra --
   240         An Approach via Module Theory'' *}
   241 lemma (in maximalideal) quotient_is_field:
   242   assumes "cring R"
   243   shows "field (R Quot I)"
   244 proof -
   245   interpret cring R by fact
   246   show ?thesis apply (intro cring.cring_fieldI2)
   247   apply (rule quotient_is_cring, rule is_cring)
   248  defer 1
   249  apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
   250  apply (simp add: rcoset_mult_add) defer 1
   251 proof (rule ccontr, simp)
   252   --{* Quotient is not empty *}
   253   assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
   254   hence II1: "I = I +> \<one>" by (simp add: FactRing_def)
   255   from a_rcos_self[OF one_closed]
   256   have "\<one> \<in> I" by (simp add: II1[symmetric])
   257   hence "I = carrier R" by (rule one_imp_carrier)
   258   from this and I_notcarr
   259   show "False" by simp
   260 next
   261   --{* Existence of Inverse *}
   262   fix a
   263   assume IanI: "I +> a \<noteq> I"
   264     and acarr: "a \<in> carrier R"
   265 
   266   --{* Helper ideal @{text "J"} *}
   267   def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
   268   have idealJ: "ideal J R"
   269     apply (unfold J_def, rule add_ideals)
   270      apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
   271     apply (rule is_ideal)
   272     done
   273 
   274   --{* Showing @{term "J"} not smaller than @{term "I"} *}
   275   have IinJ: "I \<subseteq> J"
   276   proof (rule, simp add: J_def r_coset_def set_add_defs)
   277     fix x
   278     assume xI: "x \<in> I"
   279     have Zcarr: "\<zero> \<in> carrier R" by fast
   280     from xI[THEN a_Hcarr] acarr
   281     have "x = \<zero> \<otimes> a \<oplus> x" by algebra
   282 
   283     from Zcarr and xI and this
   284     show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
   285   qed
   286 
   287   --{* Showing @{term "J \<noteq> I"} *}
   288   have anI: "a \<notin> I"
   289   proof (rule ccontr, simp)
   290     assume "a \<in> I"
   291     hence "I +> a = I" by (rule a_rcos_const)
   292     from this and IanI
   293     show "False" by simp
   294   qed
   295 
   296   have aJ: "a \<in> J"
   297   proof (simp add: J_def r_coset_def set_add_defs)
   298     from acarr
   299     have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
   300     from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this
   301     show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
   302   qed
   303 
   304   from aJ and anI
   305   have JnI: "J \<noteq> I" by fast
   306 
   307   --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
   308   from idealJ and IinJ
   309   have "J = I \<or> J = carrier R"
   310   proof (rule I_maximal, unfold J_def)
   311     have "carrier R #> a \<subseteq> carrier R"
   312       using subset_refl acarr
   313       by (rule r_coset_subset_G)
   314     from this and a_subset
   315     show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed)
   316   qed
   317 
   318   from this and JnI
   319   have Jcarr: "J = carrier R" by simp
   320 
   321   --{* Calculating an inverse for @{term "a"} *}
   322   from one_closed[folded Jcarr]
   323   have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
   324     by (simp add: J_def r_coset_def set_add_defs)
   325   from this
   326   obtain r i
   327     where rcarr: "r \<in> carrier R"
   328       and iI: "i \<in> I"
   329       and one: "\<one> = r \<otimes> a \<oplus> i"
   330     by fast
   331   from one and rcarr and acarr and iI[THEN a_Hcarr]
   332   have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
   333 
   334   --{* Lifting to cosets *}
   335   from iI
   336   have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
   337     by (intro a_rcosI, simp, intro a_subset, simp)
   338   from this and rai1
   339   have "a \<otimes> r \<in> I +> \<one>" by simp
   340   from this have "I +> \<one> = I +> a \<otimes> r"
   341     by (rule a_repr_independence, simp) (rule a_subgroup)
   342 
   343   from rcarr and this[symmetric]
   344   show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
   345 qed
   346 qed
   347 
   348 end