(* 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 _:] _ \<Otimes>\<index> _" [81,81,81] 80)

where "rcoset_mult R I A B = (\<Union>a∈A. \<Union>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) \<Otimes> (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⇙}\<one>⇘_{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 +> \<one> = I"

then have "\<one> ∈ 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 "\<zero>⇘_{R Quot I⇙}= \<one>⇘_{R Quot I⇙}"

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

from a_rcos_self[OF one_closed] have "\<one> ∈ 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: "\<zero> ∈ carrier R" by fast

from xI[THEN a_Hcarr] acarr

have "x = \<zero> ⊗ 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 = \<one> ⊗ a ⊕ \<zero>" 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. \<one> = 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: "\<one> = r ⊗ a ⊕ i" by fast

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

have rai1: "a ⊗ r = \<ominus>i ⊕ \<one>" by algebra

--{* Lifting to cosets *}

from iI have "\<ominus>i ⊕ \<one> ∈ I +> \<one>"

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

with rai1 have "a ⊗ r ∈ I +> \<one>" by simp

then have "I +> \<one> = 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 +> \<one>" by fast

qed

qed

end