(*
Title: HOL/Algebra/QuotRing.thy
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] \ 'a set"
("[mod _:] _ \\ _" [81,81,81] 80)
"rcoset_mult R I A B \ \a\A. \b\B. I +> (a \ 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)
from this 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)
from this
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)
from this
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)
from this and z2
have "\h\I. z = h \ x \ y" by fast
thus "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)
from xself and yself and zrcos
show "\a\I +> x. \b\I +> y. z \ I +> a \ b" by fast
qed
subsection {* Quotient Ring Definition *}
constdefs (structure R)
FactRing :: "[('a,'b) ring_scheme, 'a set] \ ('a set) ring"
(infixl "Quot" 65)
"FactRing R I \
\carrier = a_rcosets I, mult = rcoset_mult R I, one = (I +> \), zero = I, add = set_add R\"
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)
apply (simp add: rcoset_mult_add, fast)
--{* mult @{text one_closed} *}
apply (force intro: one_closed)
--{* mult assoc *}
apply clarify
apply (simp add: rcoset_mult_add m_assoc)
--{* mult one *}
apply clarify
apply (simp add: rcoset_mult_add l_one)
apply clarify
apply (simp add: rcoset_mult_add r_one)
--{* 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 R.is_cring)
apply (rule quotient_is_cring)
apply (rule R.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:
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)
apply (simp add: rcoset_mult_add) defer 1
proof (rule ccontr, clarsimp)
assume "I +> \ = I"
hence "\ \ I" by (simp only: a_coset_join1 one_closed a_subgroup)
hence "carrier R \ 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 \ 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"
hence "I +> y = I" by (rule a_rcos_const)
from this and b
show "False" by simp
qed
from carr
have "x \ y \ I +> x \ y" by (simp add: a_rcos_self)
from this
have xyI: "x \ y \ I" by (simp add: a)
from xyI and carr
have xI: "x \ I \ y \ I" by (simp add: I_prime)
from this and ynI
have "x \ 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:
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 "\\<^bsub>R Quot I\<^esub> = \\<^bsub>R Quot I\<^esub>"
hence II1: "I = I +> \" by (simp add: FactRing_def)
from a_rcos_self[OF one_closed]
have "\ \ 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 \ 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: "\ \ carrier R" by fast
from xI[THEN a_Hcarr] acarr
have "x = \ \ a \ x" by algebra
from Zcarr and xI and this
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"
hence "I +> a = I" by (rule a_rcos_const)
from this and 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 = \ \ a \ \" by algebra
from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this
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)
from this and a_subset
show "carrier R #> a <+> I \ 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 "\r\carrier R. \i\I. \ = r \ a \ i"
by (simp add: J_def r_coset_def set_add_defs)
from this
obtain r i
where rcarr: "r \ carrier R"
and iI: "i \ I"
and one: "\ = r \ a \ i"
by fast
from one and rcarr and acarr and iI[THEN a_Hcarr]
have rai1: "a \ r = \i \ \" by algebra
--{* Lifting to cosets *}
from iI
have "\i \ \ \ I +> \"
by (intro a_rcosI, simp, intro a_subset, simp)
from this and rai1
have "a \ r \ I +> \" by simp
from this have "I +> \ = 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 +> \" by fast
qed
qed
end