(* Title: HOL/Algebra/QuotRing.thy Author: Stephan Hohe *) theory QuotRing imports RingHom begin section ‹Quotient Rings› subsection ‹Multiplication on Cosets› definition rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] ⇒ 'a set" ("[mod _:] _ ⨂ı _" [81,81,81] 80) where "rcoset_mult R I A B = (⋃a∈A. ⋃b∈B. I +>⇘_{R⇙}(a ⊗⇘_{R⇙}b))" text ‹@{const "rcoset_mult"} fulfils the properties required by congruences› lemma (in ideal) rcoset_mult_add: "x ∈ carrier R ⟹ y ∈ carrier R ⟹ [mod I:] (I +> x) ⨂ (I +> y) = I +> (x ⊗ y)" apply rule apply (rule, simp add: rcoset_mult_def, clarsimp) defer 1 apply (rule, simp add: rcoset_mult_def) defer 1 proof - fix z x' y' assume carr: "x ∈ carrier R" "y ∈ carrier R" and x'rcos: "x' ∈ I +> x" and y'rcos: "y' ∈ I +> y" and zrcos: "z ∈ I +> x' ⊗ y'" from x'rcos have "∃h∈I. x' = h ⊕ x" by (simp add: a_r_coset_def r_coset_def) then obtain hx where hxI: "hx ∈ I" and x': "x' = hx ⊕ x" by fast+ from y'rcos have "∃h∈I. y' = h ⊕ y" by (simp add: a_r_coset_def r_coset_def) then obtain hy where hyI: "hy ∈ I" and y': "y' = hy ⊕ y" by fast+ from zrcos have "∃h∈I. z = h ⊕ (x' ⊗ y')" by (simp add: a_r_coset_def r_coset_def) then obtain hz where hzI: "hz ∈ I" and z: "z = hz ⊕ (x' ⊗ y')" by fast+ note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr] from z have "z = hz ⊕ (x' ⊗ y')" . also from x' y' have "… = hz ⊕ ((hx ⊕ x) ⊗ (hy ⊕ y))" by simp also from carr have "… = (hz ⊕ (hx ⊗ (hy ⊕ y)) ⊕ x ⊗ hy) ⊕ x ⊗ y" by algebra finally have z2: "z = (hz ⊕ (hx ⊗ (hy ⊕ y)) ⊕ x ⊗ hy) ⊕ x ⊗ y" . from hxI hyI hzI carr have "hz ⊕ (hx ⊗ (hy ⊕ y)) ⊕ x ⊗ hy ∈ I" by (simp add: I_l_closed I_r_closed) with z2 have "∃h∈I. z = h ⊕ x ⊗ y" by fast then show "z ∈ I +> x ⊗ y" by (simp add: a_r_coset_def r_coset_def) next fix z assume xcarr: "x ∈ carrier R" and ycarr: "y ∈ carrier R" and zrcos: "z ∈ I +> x ⊗ y" from xcarr have xself: "x ∈ I +> x" by (intro a_rcos_self) from ycarr have yself: "y ∈ I +> y" by (intro a_rcos_self) show "∃a∈I +> x. ∃b∈I +> y. z ∈ I +> a ⊗ b" using xself and yself and zrcos by fast qed subsection ‹Quotient Ring Definition› definition FactRing :: "[('a,'b) ring_scheme, 'a set] ⇒ ('a set) ring" (infixl "Quot" 65) where "FactRing R I = ⦇carrier = a_rcosets⇘_{R⇙}I, mult = rcoset_mult R I, one = (I +>⇘_{R⇙}𝟭⇘_{R⇙}), zero = I, add = set_add R⦈" subsection ‹Factorization over General Ideals› text ‹The quotient is a ring› lemma (in ideal) quotient_is_ring: "ring (R Quot I)" apply (rule ringI) --‹abelian group› apply (rule comm_group_abelian_groupI) apply (simp add: FactRing_def) apply (rule a_factorgroup_is_comm_group[unfolded A_FactGroup_def']) --‹mult monoid› apply (rule monoidI) apply (simp_all add: FactRing_def A_RCOSETS_def RCOSETS_def a_r_coset_def[symmetric]) --‹mult closed› apply (clarify) apply (simp add: rcoset_mult_add, fast) --‹mult @{text one_closed}› apply force --‹mult assoc› apply clarify apply (simp add: rcoset_mult_add m_assoc) --‹mult one› apply clarify apply (simp add: rcoset_mult_add) apply clarify apply (simp add: rcoset_mult_add) --‹distr› apply clarify apply (simp add: rcoset_mult_add a_rcos_sum l_distr) apply clarify apply (simp add: rcoset_mult_add a_rcos_sum r_distr) done text ‹This is a ring homomorphism› lemma (in ideal) rcos_ring_hom: "(op +> I) ∈ ring_hom R (R Quot I)" apply (rule ring_hom_memI) apply (simp add: FactRing_def a_rcosetsI[OF a_subset]) apply (simp add: FactRing_def rcoset_mult_add) apply (simp add: FactRing_def a_rcos_sum) apply (simp add: FactRing_def) done lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) (op +> I)" apply (rule ring_hom_ringI) apply (rule is_ring, rule quotient_is_ring) apply (simp add: FactRing_def a_rcosetsI[OF a_subset]) apply (simp add: FactRing_def rcoset_mult_add) apply (simp add: FactRing_def a_rcos_sum) apply (simp add: FactRing_def) done text ‹The quotient of a cring is also commutative› lemma (in ideal) quotient_is_cring: assumes "cring R" shows "cring (R Quot I)" proof - interpret cring R by fact show ?thesis apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro) apply (rule quotient_is_ring) apply (rule ring.axioms[OF quotient_is_ring]) apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric]) apply clarify apply (simp add: rcoset_mult_add m_comm) done qed text ‹Cosets as a ring homomorphism on crings› lemma (in ideal) rcos_ring_hom_cring: assumes "cring R" shows "ring_hom_cring R (R Quot I) (op +> I)" proof - interpret cring R by fact show ?thesis apply (rule ring_hom_cringI) apply (rule rcos_ring_hom_ring) apply (rule is_cring) apply (rule quotient_is_cring) apply (rule is_cring) done qed subsection ‹Factorization over Prime Ideals› text ‹The quotient ring generated by a prime ideal is a domain› lemma (in primeideal) quotient_is_domain: "domain (R Quot I)" apply (rule domain.intro) apply (rule quotient_is_cring, rule is_cring) apply (rule domain_axioms.intro) apply (simp add: FactRing_def) defer 1 apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify) apply (simp add: rcoset_mult_add) defer 1 proof (rule ccontr, clarsimp) assume "I +> 𝟭 = I" then have "𝟭 ∈ I" by (simp only: a_coset_join1 one_closed a_subgroup) then have "carrier R ⊆ I" by (subst one_imp_carrier, simp, fast) with a_subset have "I = carrier R" by fast with I_notcarr show False by fast next fix x y assume carr: "x ∈ carrier R" "y ∈ carrier R" and a: "I +> x ⊗ y = I" and b: "I +> y ≠ I" have ynI: "y ∉ I" proof (rule ccontr, simp) assume "y ∈ I" then have "I +> y = I" by (rule a_rcos_const) with b show False by simp qed from carr have "x ⊗ y ∈ I +> x ⊗ y" by (simp add: a_rcos_self) then have xyI: "x ⊗ y ∈ I" by (simp add: a) from xyI and carr have xI: "x ∈ I ∨ y ∈ I" by (simp add: I_prime) with ynI have "x ∈ I" by fast then show "I +> x = I" by (rule a_rcos_const) qed text ‹Generating right cosets of a prime ideal is a homomorphism on commutative rings› lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) (op +> I)" by (rule rcos_ring_hom_cring) (rule is_cring) subsection ‹Factorization over Maximal Ideals› text ‹In a commutative ring, the quotient ring over a maximal ideal is a field. The proof follows ``W. Adkins, S. Weintraub: Algebra -- An Approach via Module Theory''› lemma (in maximalideal) quotient_is_field: assumes "cring R" shows "field (R Quot I)" proof - interpret cring R by fact show ?thesis apply (intro cring.cring_fieldI2) apply (rule quotient_is_cring, rule is_cring) defer 1 apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp) apply (simp add: rcoset_mult_add) defer 1 proof (rule ccontr, simp) --‹Quotient is not empty› assume "𝟬⇘_{R Quot I⇙}= 𝟭⇘_{R Quot I⇙}" then have II1: "I = I +> 𝟭" by (simp add: FactRing_def) from a_rcos_self[OF one_closed] have "𝟭 ∈ I" by (simp add: II1[symmetric]) then have "I = carrier R" by (rule one_imp_carrier) with I_notcarr show False by simp next --‹Existence of Inverse› fix a assume IanI: "I +> a ≠ I" and acarr: "a ∈ carrier R" --‹Helper ideal @{text "J"}› def J ≡ "(carrier R #> a) <+> I :: 'a set" have idealJ: "ideal J R" apply (unfold J_def, rule add_ideals) apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr) apply (rule is_ideal) done --‹Showing @{term "J"} not smaller than @{term "I"}› have IinJ: "I ⊆ J" proof (rule, simp add: J_def r_coset_def set_add_defs) fix x assume xI: "x ∈ I" have Zcarr: "𝟬 ∈ carrier R" by fast from xI[THEN a_Hcarr] acarr have "x = 𝟬 ⊗ a ⊕ x" by algebra with Zcarr and xI show "∃xa∈carrier R. ∃k∈I. x = xa ⊗ a ⊕ k" by fast qed --‹Showing @{term "J ≠ I"}› have anI: "a ∉ I" proof (rule ccontr, simp) assume "a ∈ I" then have "I +> a = I" by (rule a_rcos_const) with IanI show False by simp qed have aJ: "a ∈ J" proof (simp add: J_def r_coset_def set_add_defs) from acarr have "a = 𝟭 ⊗ a ⊕ 𝟬" by algebra with one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] show "∃x∈carrier R. ∃k∈I. a = x ⊗ a ⊕ k" by fast qed from aJ and anI have JnI: "J ≠ I" by fast --‹Deducing @{term "J = carrier R"} because @{term "I"} is maximal› from idealJ and IinJ have "J = I ∨ J = carrier R" proof (rule I_maximal, unfold J_def) have "carrier R #> a ⊆ carrier R" using subset_refl acarr by (rule r_coset_subset_G) then show "carrier R #> a <+> I ⊆ carrier R" using a_subset by (rule set_add_closed) qed with JnI have Jcarr: "J = carrier R" by simp --‹Calculating an inverse for @{term "a"}› from one_closed[folded Jcarr] have "∃r∈carrier R. ∃i∈I. 𝟭 = r ⊗ a ⊕ i" by (simp add: J_def r_coset_def set_add_defs) then obtain r i where rcarr: "r ∈ carrier R" and iI: "i ∈ I" and one: "𝟭 = r ⊗ a ⊕ i" by fast from one and rcarr and acarr and iI[THEN a_Hcarr] have rai1: "a ⊗ r = ⊖i ⊕ 𝟭" by algebra --‹Lifting to cosets› from iI have "⊖i ⊕ 𝟭 ∈ I +> 𝟭" by (intro a_rcosI, simp, intro a_subset, simp) with rai1 have "a ⊗ r ∈ I +> 𝟭" by simp then have "I +> 𝟭 = I +> a ⊗ r" by (rule a_repr_independence, simp) (rule a_subgroup) from rcarr and this[symmetric] show "∃r∈carrier R. I +> a ⊗ r = I +> 𝟭" by fast qed qed end