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.
```
(*
Title:     HOL/Algebra/QuotRing.thy
Id:        \$Id\$
Author:    Stephan Hohe
*)

theory QuotRing
imports RingHom
begin

section {* Quotient Rings *}

subsection {* Multiplication on Cosets *}

constdefs (structure R)
rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"   ("[mod _:] _ \<Otimes>\<index> _" [81,81,81] 80)
"rcoset_mult R I A B \<equiv> \<Union>a\<in>A. \<Union>b\<in>B. I +> (a \<otimes> b)"

text {* @{const "rcoset_mult"} fulfils the properties required by
congruences *}
lemma (in ideal) rcoset_mult_add:
"\<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> 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 \<in> carrier R" "y \<in> carrier R"
and x'rcos: "x' \<in> I +> x"
and y'rcos: "y' \<in> I +> y"
and zrcos: "z \<in> I +> x' \<otimes> y'"

from x'rcos
have "\<exists>h\<in>I. x' = h \<oplus> x" by (simp add: a_r_coset_def r_coset_def)
from this obtain hx
where hxI: "hx \<in> I"
and x': "x' = hx \<oplus> x"
by fast+

from y'rcos
have "\<exists>h\<in>I. y' = h \<oplus> y" by (simp add: a_r_coset_def r_coset_def)
from this
obtain hy
where hyI: "hy \<in> I"
and y': "y' = hy \<oplus> y"
by fast+

from zrcos
have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')" by (simp add: a_r_coset_def r_coset_def)
from this
obtain hz
where hzI: "hz \<in> I"
and z: "z = hz \<oplus> (x' \<otimes> y')"
by fast+

note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]

from z have "z = hz \<oplus> (x' \<otimes> y')" .
also from x' y'
have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
also from carr
have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
finally
have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .

from hxI hyI hzI carr
have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"  by (simp add: I_l_closed I_r_closed)

from this and z2
have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
thus "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
next
fix z
assume xcarr: "x \<in> carrier R"
and ycarr: "y \<in> carrier R"
and zrcos: "z \<in> I +> x \<otimes> y"
from xcarr
have xself: "x \<in> I +> x" by (intro a_rcos_self)
from ycarr
have yself: "y \<in> I +> y" by (intro a_rcos_self)

from xself and yself and zrcos
show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" by fast
qed

subsection {* Quotient Ring Definition *}

constdefs (structure R)
FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
(infixl "Quot" 65)
"FactRing R I \<equiv>
\<lparr>carrier = a_rcosets I, mult = rcoset_mult R I, one = (I +> \<one>), zero = I, add = set_add R\<rparr>"

subsection {* Factorization over General Ideals *}

text {* The quotient is a ring *}
lemma (in ideal) quotient_is_ring:
shows "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)
--{* mult one\_closed *}
apply (force intro: one_closed)
--{* mult assoc *}
apply clarify
--{* mult one *}
apply clarify
apply clarify
--{* distr *}
apply clarify
apply clarify
done

text {* This is a ring homomorphism *}

lemma (in ideal) rcos_ring_hom:
"(op +> I) \<in> 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 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 a_rcos_sum)
apply (simp add: FactRing_def)
done

text {* The quotient of a cring is also commutative *}
lemma (in ideal) quotient_is_cring:
includes cring
shows "cring (R Quot I)"
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
done

text {* Cosets as a ring homomorphism on crings *}
lemma (in ideal) rcos_ring_hom_cring:
includes cring
shows "ring_hom_cring R (R Quot I) (op +> I)"
apply (rule ring_hom_cringI)
apply (rule rcos_ring_hom_ring)
apply assumption
apply (rule quotient_is_cring, assumption)
done

subsection {* Factorization over Prime Ideals *}

text {* The quotient ring generated by a prime ideal is a domain *}
lemma (in primeideal) quotient_is_domain:
shows "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)
proof (rule ccontr, clarsimp)
assume "I +> \<one> = I"
hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
from this and a_subset
have "I = carrier R" by fast
from this and I_notcarr
show "False" by fast
next
fix x y
assume carr: "x \<in> carrier R" "y \<in> carrier R"
and a: "I +> x \<otimes> y = I"
and b: "I +> y \<noteq> I"

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

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

from xyI and carr
have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
from this and ynI
have "x \<in> I" by fast
thus "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:
shows "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:
includes cring
shows "field (R Quot I)"
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)
proof (rule ccontr, simp)
--{* Quotient is not empty *}
assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
hence II1: "I = I +> \<one>" by (simp add: FactRing_def)
from a_rcos_self[OF one_closed]
have "\<one> \<in> I" by (simp add: II1[symmetric])
hence "I = carrier R" by (rule one_imp_carrier)
from this and I_notcarr
show "False" by simp
next
--{* Existence of Inverse *}
fix a
assume IanI: "I +> a \<noteq> I"
and acarr: "a \<in> carrier R"

--{* Helper ideal @{text "J"} *}
def J \<equiv> "(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 \<subseteq> J"
proof (rule, simp add: J_def r_coset_def set_add_defs)
fix x
assume xI: "x \<in> I"
have Zcarr: "\<zero> \<in> carrier R" by fast
from xI[THEN a_Hcarr] acarr
have "x = \<zero> \<otimes> a \<oplus> x" by algebra

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

--{* Showing @{term "J \<noteq> I"} *}
have anI: "a \<notin> I"
proof (rule ccontr, simp)
assume "a \<in> I"
hence "I +> a = I" by (rule a_rcos_const)
from this and IanI
show "False" by simp
qed

have aJ: "a \<in> J"
from acarr
have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this
show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
qed

from aJ and anI
have JnI: "J \<noteq> I" by fast

--{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
from idealJ and IinJ
have "J = I \<or> J = carrier R"
proof (rule I_maximal, unfold J_def)
have "carrier R #> a \<subseteq> carrier R"
by (rule r_coset_subset_G) fast
from this and a_subset
show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed)
qed

from this and JnI
have Jcarr: "J = carrier R" by simp

--{* Calculating an inverse for @{term "a"} *}
from one_closed[folded Jcarr]
have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
from this
obtain r i
where rcarr: "r \<in> carrier R"
and iI: "i \<in> I"
and one: "\<one> = r \<otimes> a \<oplus> i"
by fast
from one and rcarr and acarr and iI[THEN a_Hcarr]
have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra

--{* Lifting to cosets *}
from iI
have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
by (intro a_rcosI, simp, intro a_subset, simp)
from this and rai1
have "a \<otimes> r \<in> I +> \<one>" by simp
from this have "I +> \<one> = I +> a \<otimes> r"
by (rule a_repr_independence, simp) (rule a_subgroup)

from rcarr and this[symmetric]
show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
qed

end
```