src/HOL/Algebra/QuotRing.thy
author ballarin
Thu, 03 Aug 2006 14:57:26 +0200
changeset 20318 0e0ea63fe768
child 21502 7f3ea2b3bab6
permissions -rw-r--r--
Restructured algebra library, added ideals and quotient rings.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
20318
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(*
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  Title:     HOL/Algebra/QuotRing.thy
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  Id:        $Id$
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  Author:    Stephan Hohe
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*)
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theory QuotRing
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imports RingHom
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begin
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section {* Quotient Rings *}
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subsection {* Multiplication on Cosets *}
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constdefs (structure R)
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  rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"   ("[mod _:] _ \<Otimes>\<index> _" [81,81,81] 80)
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  "rcoset_mult R I A B \<equiv> \<Union>a\<in>A. \<Union>b\<in>B. I +> (a \<otimes> b)"
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text {* @{const "rcoset_mult"} fulfils the properties required by
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  congruences *}
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lemma (in ideal) rcoset_mult_add:
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  "\<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
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apply rule
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apply (rule, simp add: rcoset_mult_def, clarsimp)
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defer 1
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apply (rule, simp add: rcoset_mult_def)
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defer 1
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proof -
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  fix z x' y'
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  assume carr: "x \<in> carrier R" "y \<in> carrier R"
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     and x'rcos: "x' \<in> I +> x"
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     and y'rcos: "y' \<in> I +> y"
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     and zrcos: "z \<in> I +> x' \<otimes> y'"
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  from x'rcos 
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      have "\<exists>h\<in>I. x' = h \<oplus> x" by (simp add: a_r_coset_def r_coset_def)
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  from this obtain hx
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      where hxI: "hx \<in> I"
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      and x': "x' = hx \<oplus> x"
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      by fast+
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  from y'rcos
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      have "\<exists>h\<in>I. y' = h \<oplus> y" by (simp add: a_r_coset_def r_coset_def)
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  from this
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      obtain hy
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      where hyI: "hy \<in> I"
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      and y': "y' = hy \<oplus> y"
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      by fast+
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  from zrcos
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      have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')" by (simp add: a_r_coset_def r_coset_def)
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  from this
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      obtain hz
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      where hzI: "hz \<in> I"
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      and z: "z = hz \<oplus> (x' \<otimes> y')"
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      by fast+
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  note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
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  from z have "z = hz \<oplus> (x' \<otimes> y')" .
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  also from x' y'
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      have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
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  also from carr
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      have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
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  finally
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    68
      have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
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  from hxI hyI hzI carr
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    71
      have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"  by (simp add: I_l_closed I_r_closed)
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    73
  from this and z2
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    74
      have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
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  thus "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
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next
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  fix z
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  assume xcarr: "x \<in> carrier R"
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     and ycarr: "y \<in> carrier R"
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    80
     and zrcos: "z \<in> I +> x \<otimes> y"
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    81
  from xcarr
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    82
      have xself: "x \<in> I +> x" by (intro a_rcos_self)
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    83
  from ycarr
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    84
      have yself: "y \<in> I +> y" by (intro a_rcos_self)
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    86
  from xself and yself and zrcos
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    87
      show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" by fast
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qed
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    91
subsection {* Quotient Ring Definition *}
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constdefs (structure R)
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    94
  FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
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    95
     (infixl "Quot" 65)
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    96
  "FactRing R I \<equiv>
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    \<lparr>carrier = a_rcosets I, mult = rcoset_mult R I, one = (I +> \<one>), zero = I, add = set_add R\<rparr>"
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   100
subsection {* Factorization over General Ideals *}
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   101
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text {* The quotient is a ring *}
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   103
lemma (in ideal) quotient_is_ring:
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   104
  shows "ring (R Quot I)"
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   105
apply (rule ringI)
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   --{* abelian group *}
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   107
   apply (rule comm_group_abelian_groupI)
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   108
   apply (simp add: FactRing_def)
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   109
   apply (rule a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
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   110
  --{* mult monoid *}
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   111
  apply (rule monoidI)
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   112
      apply (simp_all add: FactRing_def A_RCOSETS_def RCOSETS_def
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   113
             a_r_coset_def[symmetric])
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   114
      --{* mult closed *}
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   115
      apply (clarify)
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   116
      apply (simp add: rcoset_mult_add, fast)
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   117
     --{* mult one\_closed *}
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   118
     apply (force intro: one_closed)
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   119
    --{* mult assoc *}
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   120
    apply clarify
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   121
    apply (simp add: rcoset_mult_add m_assoc)
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   122
   --{* mult one *}
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   123
   apply clarify
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   124
   apply (simp add: rcoset_mult_add l_one)
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   125
  apply clarify
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   126
  apply (simp add: rcoset_mult_add r_one)
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   127
 --{* distr *}
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   128
 apply clarify
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   129
 apply (simp add: rcoset_mult_add a_rcos_sum l_distr)
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   130
apply clarify
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   131
apply (simp add: rcoset_mult_add a_rcos_sum r_distr)
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   132
done
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   133
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   134
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text {* This is a ring homomorphism *}
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   136
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   137
lemma (in ideal) rcos_ring_hom:
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   138
  "(op +> I) \<in> ring_hom R (R Quot I)"
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   139
apply (rule ring_hom_memI)
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   140
   apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
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   141
  apply (simp add: FactRing_def rcoset_mult_add)
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   142
 apply (simp add: FactRing_def a_rcos_sum)
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   143
apply (simp add: FactRing_def)
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   144
done
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   145
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   146
lemma (in ideal) rcos_ring_hom_ring:
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   147
  "ring_hom_ring R (R Quot I) (op +> I)"
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   148
apply (rule ring_hom_ringI)
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   149
     apply (rule is_ring, rule quotient_is_ring)
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   150
   apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
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   151
  apply (simp add: FactRing_def rcoset_mult_add)
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   152
 apply (simp add: FactRing_def a_rcos_sum)
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   153
apply (simp add: FactRing_def)
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   154
done
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   155
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ballarin
parents:
diff changeset
   156
text {* The quotient of a cring is also commutative *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   157
lemma (in ideal) quotient_is_cring:
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   158
  includes cring
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   159
  shows "cring (R Quot I)"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   160
apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   161
  apply (rule quotient_is_ring)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   162
 apply (rule ring.axioms[OF quotient_is_ring])
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   163
apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   164
apply clarify
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   165
apply (simp add: rcoset_mult_add m_comm)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   166
done
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   167
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   168
text {* Cosets as a ring homomorphism on crings *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   169
lemma (in ideal) rcos_ring_hom_cring:
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   170
  includes cring
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   171
  shows "ring_hom_cring R (R Quot I) (op +> I)"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   172
apply (rule ring_hom_cringI)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   173
  apply (rule rcos_ring_hom_ring)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   174
 apply assumption
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   175
apply (rule quotient_is_cring, assumption)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   176
done
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   177
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   178
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   179
subsection {* Factorization over Prime Ideals *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   180
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   181
text {* The quotient ring generated by a prime ideal is a domain *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   182
lemma (in primeideal) quotient_is_domain:
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   183
  shows "domain (R Quot I)"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   184
apply (rule domain.intro)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   185
 apply (rule quotient_is_cring, rule is_cring)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   186
apply (rule domain_axioms.intro)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   187
 apply (simp add: FactRing_def) defer 1
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   188
 apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   189
 apply (simp add: rcoset_mult_add) defer 1
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   190
proof (rule ccontr, clarsimp)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   191
  assume "I +> \<one> = I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   192
  hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   193
  hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   194
  from this and a_subset
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   195
      have "I = carrier R" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   196
  from this and I_notcarr
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   197
      show "False" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   198
next
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   199
  fix x y
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   200
  assume carr: "x \<in> carrier R" "y \<in> carrier R"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   201
     and a: "I +> x \<otimes> y = I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   202
     and b: "I +> y \<noteq> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   203
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   204
  have ynI: "y \<notin> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   205
  proof (rule ccontr, simp)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   206
    assume "y \<in> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   207
    hence "I +> y = I" by (rule a_rcos_const)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   208
    from this and b
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   209
        show "False" by simp
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   210
  qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   211
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   212
  from carr
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   213
      have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   214
  from this
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   215
      have xyI: "x \<otimes> y \<in> I" by (simp add: a)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   216
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   217
  from xyI and carr
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   218
      have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   219
  from this and ynI
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   220
      have "x \<in> I" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   221
  thus "I +> x = I" by (rule a_rcos_const)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   222
qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   223
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   224
text {* Generating right cosets of a prime ideal is a homomorphism
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   225
        on commutative rings *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   226
lemma (in primeideal) rcos_ring_hom_cring:
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   227
  shows "ring_hom_cring R (R Quot I) (op +> I)"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   228
by (rule rcos_ring_hom_cring, rule is_cring)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   229
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   230
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   231
subsection {* Factorization over Maximal Ideals *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   232
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   233
text {* In a commutative ring, the quotient ring over a maximal ideal
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   234
        is a field.
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   235
        The proof follows ``W. Adkins, S. Weintraub: Algebra --
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   236
        An Approach via Module Theory'' *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   237
lemma (in maximalideal) quotient_is_field:
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   238
  includes cring
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   239
  shows "field (R Quot I)"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   240
apply (intro cring.cring_fieldI2)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   241
  apply (rule quotient_is_cring, rule is_cring)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   242
 defer 1
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   243
 apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   244
 apply (simp add: rcoset_mult_add) defer 1
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   245
proof (rule ccontr, simp)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   246
  --{* Quotient is not empty *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   247
  assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   248
  hence II1: "I = I +> \<one>" by (simp add: FactRing_def)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   249
  from a_rcos_self[OF one_closed]
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   250
      have "\<one> \<in> I" by (simp add: II1[symmetric])
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   251
  hence "I = carrier R" by (rule one_imp_carrier)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   252
  from this and I_notcarr
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   253
      show "False" by simp
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   254
next
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   255
  --{* Existence of Inverse *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   256
  fix a
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   257
  assume IanI: "I +> a \<noteq> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   258
     and acarr: "a \<in> carrier R"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   259
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   260
  --{* Helper ideal @{text "J"} *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   261
  def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   262
  have idealJ: "ideal J R"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   263
      apply (unfold J_def, rule add_ideals)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   264
      apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   265
      apply (rule is_ideal)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   266
      done
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   267
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   268
  --{* Showing @{term "J"} not smaller than @{term "I"} *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   269
  have IinJ: "I \<subseteq> J"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   270
  proof (rule, simp add: J_def r_coset_def set_add_defs)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   271
    fix x
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   272
    assume xI: "x \<in> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   273
    have Zcarr: "\<zero> \<in> carrier R" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   274
    from xI[THEN a_Hcarr] acarr
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   275
    have "x = \<zero> \<otimes> a \<oplus> x" by algebra
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   276
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   277
    from Zcarr and xI and this
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   278
        show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   279
  qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   280
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   281
  --{* Showing @{term "J \<noteq> I"} *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   282
  have anI: "a \<notin> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   283
  proof (rule ccontr, simp)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   284
    assume "a \<in> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   285
    hence "I +> a = I" by (rule a_rcos_const)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   286
    from this and IanI
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   287
        show "False" by simp
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   288
  qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   289
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   290
  have aJ: "a \<in> J"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   291
  proof (simp add: J_def r_coset_def set_add_defs)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   292
    from acarr
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   293
        have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   294
    from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   295
        show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   296
  qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   297
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   298
  from aJ and anI
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   299
      have JnI: "J \<noteq> I" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   300
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   301
  --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   302
  from idealJ and IinJ
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   303
      have "J = I \<or> J = carrier R"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   304
  proof (rule I_maximal, unfold J_def)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   305
    have "carrier R #> a \<subseteq> carrier R"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   306
	  by (rule r_coset_subset_G) fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   307
    from this and a_subset
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   308
        show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   309
  qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   310
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   311
  from this and JnI
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   312
      have Jcarr: "J = carrier R" by simp
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   313
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   314
  --{* Calculating an inverse for @{term "a"} *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   315
  from one_closed[folded Jcarr]
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   316
      have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   317
      by (simp add: J_def r_coset_def set_add_defs)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   318
  from this
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   319
     obtain r i
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   320
       where rcarr: "r \<in> carrier R"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   321
       and iI: "i \<in> I"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   322
       and one: "\<one> = r \<otimes> a \<oplus> i"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   323
     by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   324
  from one and rcarr and acarr and iI[THEN a_Hcarr]
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   325
      have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   326
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   327
  --{* Lifting to cosets *}
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   328
  from iI
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   329
      have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   330
      by (intro a_rcosI, simp, intro a_subset, simp)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   331
  from this and rai1
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   332
      have "a \<otimes> r \<in> I +> \<one>" by simp
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   333
  from this have "I +> \<one> = I +> a \<otimes> r"
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   334
      by (rule a_repr_independence, simp) (rule a_subgroup)
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   335
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   336
  from rcarr and this[symmetric]
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   337
      show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   338
qed
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   339
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff changeset
   340
end