author  wenzelm 
Mon, 19 Sep 2011 23:18:18 +0200  
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parent 35849  b5522b51cb1e 
child 61382  efac889fccbc 
permissions  rwrr 
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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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section {* Quotient Rings *} 
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subsection {* Multiplication on Cosets *} 
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definition 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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where "rcoset_mult R I A B = (\<Union>a\<in>A. \<Union>b\<in>B. I +>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> 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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"x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<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 have "\<exists>h\<in>I. x' = h \<oplus> x" 

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by (simp add: a_r_coset_def r_coset_def) 

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then obtain hx where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x" 

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by fast+ 

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from y'rcos have "\<exists>h\<in>I. y' = h \<oplus> y" 
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by (simp add: a_r_coset_def r_coset_def) 

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then obtain hy where hyI: "hy \<in> I" and y': "y' = hy \<oplus> y" 

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by fast+ 

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from zrcos have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')" 
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by (simp add: a_r_coset_def r_coset_def) 

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then obtain hz where hzI: "hz \<in> I" 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' have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp 
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also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra 

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finally 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 have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I" 
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by (simp add: I_l_closed I_r_closed) 

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with z2 have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast 
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then show "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 have xself: "x \<in> I +> x" by (intro a_rcos_self) 

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from ycarr have yself: "y \<in> I +> y" by (intro a_rcos_self) 

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show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" 

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using xself and yself and zrcos by fast 

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qed 
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subsection {* Quotient Ring Definition *} 
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definition FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring" 
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(infixl "Quot" 65) 

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where "FactRing R I = 
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\<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, 
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one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), 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: "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) 
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{* mult @{text one_closed} *} 
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apply force 
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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) 
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apply clarify 
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apply (simp add: rcoset_mult_add) 
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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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lemma (in ideal) rcos_ring_hom: "(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: "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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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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text {* The quotient of a cring is also commutative *} 
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lemma (in ideal) quotient_is_cring: 
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assumes "cring R" 
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shows "cring (R Quot I)" 
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proof  
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interpret cring R by fact 
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show ?thesis 
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apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro) 

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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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done 

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qed 
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text {* Cosets as a ring homomorphism on crings *} 
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lemma (in ideal) rcos_ring_hom_cring: 
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assumes "cring R" 
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shows "ring_hom_cring R (R Quot I) (op +> I)" 
27611  155 
proof  
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interpret cring R by fact 
45005  157 
show ?thesis 
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apply (rule ring_hom_cringI) 

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apply (rule rcos_ring_hom_ring) 

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apply (rule is_cring) 

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apply (rule quotient_is_cring) 

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apply (rule is_cring) 

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done 

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qed 
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subsection {* Factorization over Prime Ideals *} 
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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: "domain (R Quot I)" 
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apply (rule domain.intro) 

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apply (rule quotient_is_cring, rule is_cring) 

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apply (rule domain_axioms.intro) 

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apply (simp add: FactRing_def) defer 1 

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apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify) 

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apply (simp add: rcoset_mult_add) defer 1 

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proof (rule ccontr, clarsimp) 
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assume "I +> \<one> = I" 
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then have "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup) 
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then have "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast) 

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with a_subset have "I = carrier R" by fast 

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with I_notcarr show False by fast 

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next 
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fix x y 
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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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and b: "I +> y \<noteq> I" 

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have ynI: "y \<notin> I" 
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proof (rule ccontr, simp) 
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assume "y \<in> I" 
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then have "I +> y = I" by (rule a_rcos_const) 
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with b show False by simp 

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qed 
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from carr have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self) 
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then have xyI: "x \<otimes> y \<in> I" by (simp add: a) 

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from xyI and carr have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime) 
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with ynI have "x \<in> I" by fast 

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then show "I +> x = I" by (rule a_rcos_const) 

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

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subsection {* Factorization over Maximal Ideals *} 
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text {* In a commutative ring, the quotient ring over a maximal ideal 
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is a field. 
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The proof follows ``W. Adkins, S. Weintraub: Algebra  
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An Approach via Module Theory'' *} 
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lemma (in maximalideal) quotient_is_field: 
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assumes "cring R" 
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shows "field (R Quot I)" 
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proof  
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interpret cring R by fact 
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show ?thesis 
222 
apply (intro cring.cring_fieldI2) 

223 
apply (rule quotient_is_cring, rule is_cring) 

224 
defer 1 

225 
apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp) 

226 
apply (simp add: rcoset_mult_add) defer 1 

227 
proof (rule ccontr, simp) 

228 
{* Quotient is not empty *} 

229 
assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>" 

230 
then have II1: "I = I +> \<one>" by (simp add: FactRing_def) 

231 
from a_rcos_self[OF one_closed] have "\<one> \<in> I" 

232 
by (simp add: II1[symmetric]) 

233 
then have "I = carrier R" by (rule one_imp_carrier) 

234 
with I_notcarr show False by simp 

235 
next 

236 
{* Existence of Inverse *} 

237 
fix a 

238 
assume IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R" 

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{* Helper ideal @{text "J"} *} 
241 
def J \<equiv> "(carrier R #> a) <+> I :: 'a set" 

242 
have idealJ: "ideal J R" 

243 
apply (unfold J_def, rule add_ideals) 

244 
apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr) 

245 
apply (rule is_ideal) 

246 
done 

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{* Showing @{term "J"} not smaller than @{term "I"} *} 
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have IinJ: "I \<subseteq> J" 

250 
proof (rule, simp add: J_def r_coset_def set_add_defs) 

251 
fix x 

252 
assume xI: "x \<in> I" 

253 
have Zcarr: "\<zero> \<in> carrier R" by fast 

254 
from xI[THEN a_Hcarr] acarr 

255 
have "x = \<zero> \<otimes> a \<oplus> x" by algebra 

256 
with Zcarr and xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast 

257 
qed 

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{* Showing @{term "J \<noteq> I"} *} 
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have anI: "a \<notin> I" 

261 
proof (rule ccontr, simp) 

262 
assume "a \<in> I" 

263 
then have "I +> a = I" by (rule a_rcos_const) 

264 
with IanI show False by simp 

265 
qed 

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45005  267 
have aJ: "a \<in> J" 
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proof (simp add: J_def r_coset_def set_add_defs) 

269 
from acarr 

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have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra 

271 
with one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] 

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show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast 

273 
qed 

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from aJ and anI have JnI: "J \<noteq> I" by fast 
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{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *} 
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from idealJ and IinJ have "J = I \<or> J = carrier R" 

279 
proof (rule I_maximal, unfold J_def) 

280 
have "carrier R #> a \<subseteq> carrier R" 

281 
using subset_refl acarr by (rule r_coset_subset_G) 

282 
then show "carrier R #> a <+> I \<subseteq> carrier R" 

283 
using a_subset by (rule set_add_closed) 

284 
qed 

285 

286 
with JnI have Jcarr: "J = carrier R" by simp 

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45005  288 
{* Calculating an inverse for @{term "a"} *} 
289 
from one_closed[folded Jcarr] 

290 
have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i" 

291 
by (simp add: J_def r_coset_def set_add_defs) 

292 
then obtain r i where rcarr: "r \<in> carrier R" 

293 
and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i" by fast 

294 
from one and rcarr and acarr and iI[THEN a_Hcarr] 

295 
have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra 

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45005  297 
{* Lifting to cosets *} 
298 
from iI have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>" 

299 
by (intro a_rcosI, simp, intro a_subset, simp) 

300 
with rai1 have "a \<otimes> r \<in> I +> \<one>" by simp 

301 
then have "I +> \<one> = I +> a \<otimes> r" 

302 
by (rule a_repr_independence, simp) (rule a_subgroup) 

303 

304 
from rcarr and this[symmetric] 

305 
show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast 

306 
qed 

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qed 
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end 