author  ballarin 
Fri, 19 Dec 2008 11:09:09 +0100  
changeset 29242  e190bc2a5399 
parent 29237  e90d9d51106b 
child 35847  19f1f7066917 
permissions  rwrr 
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(* 
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Title: HOL/Algebra/QuotRing.thy 
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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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Generalised polynomial lemmas from cring to ring.
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section {* Quotient Rings *} 
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Generalised polynomial lemmas from cring to ring.
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subsection {* Multiplication on Cosets *} 
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constdefs (structure R) 
23463  15 
rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set" 
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("[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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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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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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from this and z2 
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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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and zrcos: "z \<in> I +> x \<otimes> y" 
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from xcarr 
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have xself: "x \<in> I +> x" by (intro a_rcos_self) 
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from ycarr 
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have yself: "y \<in> I +> y" by (intro a_rcos_self) 
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from xself and yself and zrcos 
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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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subsection {* Quotient Ring Definition *} 
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constdefs (structure R) 
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FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring" 
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(infixl "Quot" 65) 
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"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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subsection {* Factorization over General Ideals *} 
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text {* The quotient is a ring *} 
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lemma (in ideal) quotient_is_ring: 
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shows "ring (R Quot I)" 
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apply (rule ringI) 
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{* abelian group *} 
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apply (rule comm_group_abelian_groupI) 
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apply (simp add: FactRing_def) 
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apply (rule a_factorgroup_is_comm_group[unfolded A_FactGroup_def']) 
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{* mult monoid *} 
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apply (rule monoidI) 
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apply (simp_all add: FactRing_def A_RCOSETS_def RCOSETS_def 
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a_r_coset_def[symmetric]) 
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{* mult closed *} 
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apply (clarify) 
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apply (simp add: rcoset_mult_add, fast) 
21502  116 
{* mult @{text one_closed} *} 
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apply (force intro: one_closed) 
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{* mult assoc *} 
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apply clarify 
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apply (simp add: rcoset_mult_add m_assoc) 
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{* mult one *} 
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apply clarify 
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apply (simp add: rcoset_mult_add l_one) 
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apply clarify 
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apply (simp add: rcoset_mult_add r_one) 
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{* distr *} 
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apply clarify 
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apply (simp add: rcoset_mult_add a_rcos_sum l_distr) 
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apply clarify 
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apply (simp add: rcoset_mult_add a_rcos_sum r_distr) 
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done 
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text {* This is a ring homomorphism *} 
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135 

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lemma (in ideal) rcos_ring_hom: 
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"(op +> I) \<in> ring_hom R (R Quot I)" 
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apply (rule ring_hom_memI) 
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apply (simp add: FactRing_def a_rcosetsI[OF a_subset]) 
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apply (simp add: FactRing_def rcoset_mult_add) 
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apply (simp add: FactRing_def a_rcos_sum) 
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apply (simp add: FactRing_def) 
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done 
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lemma (in ideal) rcos_ring_hom_ring: 
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"ring_hom_ring R (R Quot I) (op +> I)" 
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apply (rule ring_hom_ringI) 
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apply (rule is_ring, rule quotient_is_ring) 
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apply (simp add: FactRing_def a_rcosetsI[OF a_subset]) 
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apply (simp add: FactRing_def rcoset_mult_add) 
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151 
apply (simp add: FactRing_def a_rcos_sum) 
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apply (simp add: FactRing_def) 
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done 
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154 

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text {* The quotient of a cring is also commutative *} 
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lemma (in ideal) quotient_is_cring: 
27611  157 
assumes "cring R" 
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158 
shows "cring (R Quot I)" 
27611  159 
proof  
29237  160 
interpret cring R by fact 
27611  161 
show ?thesis apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro) 
20318
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162 
apply (rule quotient_is_ring) 
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apply (rule ring.axioms[OF quotient_is_ring]) 
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apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric]) 
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apply clarify 
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apply (simp add: rcoset_mult_add m_comm) 
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167 
done 
27611  168 
qed 
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169 

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text {* Cosets as a ring homomorphism on crings *} 
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lemma (in ideal) rcos_ring_hom_cring: 
27611  172 
assumes "cring R" 
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173 
shows "ring_hom_cring R (R Quot I) (op +> I)" 
27611  174 
proof  
29237  175 
interpret cring R by fact 
27611  176 
show ?thesis apply (rule ring_hom_cringI) 
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177 
apply (rule rcos_ring_hom_ring) 
29242  178 
apply (rule is_cring) 
23463  179 
apply (rule quotient_is_cring) 
29242  180 
apply (rule is_cring) 
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181 
done 
27611  182 
qed 
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183 

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184 
subsection {* Factorization over Prime Ideals *} 
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185 

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text {* The quotient ring generated by a prime ideal is a domain *} 
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lemma (in primeideal) quotient_is_domain: 
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shows "domain (R Quot I)" 
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189 
apply (rule domain.intro) 
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190 
apply (rule quotient_is_cring, rule is_cring) 
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191 
apply (rule domain_axioms.intro) 
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192 
apply (simp add: FactRing_def) defer 1 
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193 
apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify) 
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194 
apply (simp add: rcoset_mult_add) defer 1 
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195 
proof (rule ccontr, clarsimp) 
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196 
assume "I +> \<one> = I" 
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hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup) 
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198 
hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast) 
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199 
from this and a_subset 
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200 
have "I = carrier R" by fast 
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201 
from this and I_notcarr 
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show "False" by fast 
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203 
next 
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204 
fix x y 
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205 
assume carr: "x \<in> carrier R" "y \<in> carrier R" 
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and a: "I +> x \<otimes> y = I" 
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207 
and b: "I +> y \<noteq> I" 
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208 

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have ynI: "y \<notin> I" 
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210 
proof (rule ccontr, simp) 
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211 
assume "y \<in> I" 
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212 
hence "I +> y = I" by (rule a_rcos_const) 
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213 
from this and b 
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214 
show "False" by simp 
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215 
qed 
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216 

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217 
from carr 
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218 
have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self) 
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219 
from this 
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220 
have xyI: "x \<otimes> y \<in> I" by (simp add: a) 
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221 

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222 
from xyI and carr 
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223 
have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime) 
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224 
from this and ynI 
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225 
have "x \<in> I" by fast 
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226 
thus "I +> x = I" by (rule a_rcos_const) 
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227 
qed 
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228 

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229 
text {* Generating right cosets of a prime ideal is a homomorphism 
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on commutative rings *} 
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231 
lemma (in primeideal) rcos_ring_hom_cring: 
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232 
shows "ring_hom_cring R (R Quot I) (op +> I)" 
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233 
by (rule rcos_ring_hom_cring, rule is_cring) 
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234 

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235 

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236 
subsection {* Factorization over Maximal Ideals *} 
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237 

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text {* In a commutative ring, the quotient ring over a maximal ideal 
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239 
is a field. 
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240 
The proof follows ``W. Adkins, S. Weintraub: Algebra  
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241 
An Approach via Module Theory'' *} 
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242 
lemma (in maximalideal) quotient_is_field: 
27611  243 
assumes "cring R" 
20318
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244 
shows "field (R Quot I)" 
27611  245 
proof  
29237  246 
interpret cring R by fact 
27611  247 
show ?thesis apply (intro cring.cring_fieldI2) 
20318
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248 
apply (rule quotient_is_cring, rule is_cring) 
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249 
defer 1 
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250 
apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp) 
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251 
apply (simp add: rcoset_mult_add) defer 1 
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252 
proof (rule ccontr, simp) 
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253 
{* Quotient is not empty *} 
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254 
assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>" 
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255 
hence II1: "I = I +> \<one>" by (simp add: FactRing_def) 
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256 
from a_rcos_self[OF one_closed] 
23350  257 
have "\<one> \<in> I" by (simp add: II1[symmetric]) 
20318
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258 
hence "I = carrier R" by (rule one_imp_carrier) 
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259 
from this and I_notcarr 
23350  260 
show "False" by simp 
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261 
next 
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262 
{* Existence of Inverse *} 
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263 
fix a 
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264 
assume IanI: "I +> a \<noteq> I" 
23350  265 
and acarr: "a \<in> carrier R" 
20318
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266 

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267 
{* Helper ideal @{text "J"} *} 
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268 
def J \<equiv> "(carrier R #> a) <+> I :: 'a set" 
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269 
have idealJ: "ideal J R" 
23350  270 
apply (unfold J_def, rule add_ideals) 
271 
apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr) 

272 
apply (rule is_ideal) 

273 
done 

20318
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274 

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275 
{* Showing @{term "J"} not smaller than @{term "I"} *} 
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276 
have IinJ: "I \<subseteq> J" 
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277 
proof (rule, simp add: J_def r_coset_def set_add_defs) 
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278 
fix x 
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279 
assume xI: "x \<in> I" 
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280 
have Zcarr: "\<zero> \<in> carrier R" by fast 
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281 
from xI[THEN a_Hcarr] acarr 
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282 
have "x = \<zero> \<otimes> a \<oplus> x" by algebra 
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283 

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284 
from Zcarr and xI and this 
23350  285 
show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast 
20318
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286 
qed 
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Restructured algebra library, added ideals and quotient rings.
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287 

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288 
{* Showing @{term "J \<noteq> I"} *} 
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289 
have anI: "a \<notin> I" 
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290 
proof (rule ccontr, simp) 
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291 
assume "a \<in> I" 
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292 
hence "I +> a = I" by (rule a_rcos_const) 
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293 
from this and IanI 
23350  294 
show "False" by simp 
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295 
qed 
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296 

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297 
have aJ: "a \<in> J" 
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298 
proof (simp add: J_def r_coset_def set_add_defs) 
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299 
from acarr 
23350  300 
have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra 
20318
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301 
from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this 
23350  302 
show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast 
20318
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303 
qed 
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304 

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305 
from aJ and anI 
23350  306 
have JnI: "J \<noteq> I" by fast 
20318
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307 

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308 
{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *} 
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309 
from idealJ and IinJ 
23350  310 
have "J = I \<or> J = carrier R" 
20318
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311 
proof (rule I_maximal, unfold J_def) 
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312 
have "carrier R #> a \<subseteq> carrier R" 
23350  313 
using subset_refl acarr 
314 
by (rule r_coset_subset_G) 

20318
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315 
from this and a_subset 
23350  316 
show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed) 
20318
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317 
qed 
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318 

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319 
from this and JnI 
23350  320 
have Jcarr: "J = carrier R" by simp 
20318
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321 

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322 
{* Calculating an inverse for @{term "a"} *} 
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323 
from one_closed[folded Jcarr] 
23350  324 
have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i" 
325 
by (simp add: J_def r_coset_def set_add_defs) 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

326 
from this 
23350  327 
obtain r i 
328 
where rcarr: "r \<in> carrier R" 

329 
and iI: "i \<in> I" 

330 
and one: "\<one> = r \<otimes> a \<oplus> i" 

331 
by fast 

20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

332 
from one and rcarr and acarr and iI[THEN a_Hcarr] 
23350  333 
have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

334 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

335 
{* Lifting to cosets *} 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

336 
from iI 
23350  337 
have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>" 
338 
by (intro a_rcosI, simp, intro a_subset, simp) 

20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

339 
from this and rai1 
23350  340 
have "a \<otimes> r \<in> I +> \<one>" by simp 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

341 
from this have "I +> \<one> = I +> a \<otimes> r" 
23350  342 
by (rule a_repr_independence, simp) (rule a_subgroup) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

343 

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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

344 
from rcarr and this[symmetric] 
23350  345 
show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast 
20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

346 
qed 
27611  347 
qed 
20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

348 

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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

349 
end 