cấu trúc các ideal trong vành đa thức trên miền nguyên dedekind, chương 3

CHVdNG 3
CA"U TRUC cAC IDEAL TRONG VANH fJA THUC
TREN MIEN NGUYEN DEDEKIND
Trang ehu'dng nay, ta se m6 ta diu tfl.k cLlaideal ba't ky trang vanh c1athue
tren mi~n nguyen Dedekind D. Vi D[x] la vanh Noether nen moi ideal trang
D[x]a~u e6 s\j phan tfch nguyen sd ehu5n hoa, do d6 d~ m6 ta ea'u true et'1a
ideal ba't ky trang D[x] ta n~6ta ea'u true cae ideal nguyen sd. Khi a~ e~p den
P, I, J, ideal kh;k kh6ng eth D[x];ha't kha quy trang D[.r]!P[x].

3.1 Call true cua cae ideal nguyen sd trong D[x]
Truoe khi 1116ta ea'u tnk eua cae ideal nguyen Sd, nh~e b;d dlng, neu I la
ideal nguyen sd elh D[x] thi theo Mfnh di 1.4.5, Rad(I) nguyen to trang D[~r],

va do a6 theo ehu'dng 1, Rad(I) toi a:;tiho~e kh6ng toi a;;li. Sau day ta se 11n
IltQ11116ta d:;tngeua ideal nguyen sd I ung voi m6i Rad(l) nhu' tren.

3.1.1 Dinh Ij (M6 ta cae ideal nguyen sd voi can kh6ng toi a:;ti). Ideal I Clla
D[x] fa nguyen sO'v6i din khong tal dfli neu va chi nlu I = In v6i n nguyen
du:O'ngva J fa ideal nguyen to/khong tal dfli ala D[T].
Chllng mink.

Theo Dinh Ij 1.6.10, D[x] la Q-vanh, nen theo Dinh /j 1.6.6

111Qiideal nguyen

to kh6ng

toi a:;ti eua D[x] la ideal nhan. Hdn nlfa mQi ideal

ngllyen to kh6ng toi a:;tikhae kh6ng euaD[x] d~ll e6 so ehi~u Ian hdn kh6ng,
tLta6, dung Bo?di 1.1 trang [1]ta e6 ai~ll phai eh(tng minh.\?

3.1.2 Hf qua. Trang D[I]) m9i lug thua ala ideal nguyen to' d~u nguyen sO',
18


De m6 t~td~lng cLb ide:ll nguyen so \'6'j can t6i c.1~ti
ta dn Ixi c.12sau

3.1.3 Blf dl. Ideal I ala D[x]ta nguyen sd Val din tal'd9i neu va chi neu ton

t9i idealngllyen to'P -#0 ala D, da thuc don khdi ip(x)

E D[:c]thod ip(x)+ P[:r]

be/tkhd quy !rong D[xJ/P[x]va k, l nguyen duong sao cho k(x)]1.:E I, pi C I.
Hdn mia, khz d6 Rad(I) =< ip(x), P >.

Chung minh. (==?)Gia su I nguyenso voi can toi d~tithl Rad(I) =< rp(x),P >,
trang d6 ip(x). P nhu da quy uoc. Do ip(x) E Rad(I), P c Rad(I) Den c6 k, l
nguyen dltong sao cho [;:(x)]1.:
E I, pi C I.
(~)

Gia Slt c6 k, I nguyen dltong sao cho [E I, pi C I, thl

ip(x) E Rad(I). P c Rad(I).

Do d6, < ip(x), P >c RadJ).

Theo Bo? di 2.4,

< ip(x), P > toj d?i trang D[x]Den Rad(I) =< ip(x),P > ho?c Rad(I) = D[x].VI
I -:JD[x] Den rhea M~nh di 1.4.1 ii), Rad(I) -:JD[x], do d6 Rad(I) =< ip(x), P >
toi d?i trang D[.r]. Theo Al~nh di 1.4.6, Q nguyen so rrang D[x]. \?

3.1.4 Binh If (M6 ra c:k ideal nguyen so voi can toi d~li).Ideal I ala D[x]ta
nguyen sd v6l din tal d9i neu va chi neu I c6 dflng
< [;p(x)]m,pn, ip(x)h1(x)+ Pl(X), ..., ip(x)ht(x) -l-Pt(x) >,
!rong d6 m, n nguyen duong, t kh6ng am, hi (x) ("1::; i ::; t) thwjc D[x], Pi(x) (1 ::;
i ::; t) thu(Jc P[x]. Hon mJa, khz d6 Rad(I) =< ip(x), P > z:dta c6 thi ch9n hi(x),
Pi(x) thoa deg[hi(x)] ::; (Tn- l)deg[ip(x)], deg[Pi(x)] ::; Tndeg~;(x)].

Chung minh. (==?)Gia su I nguyenso voi can toi d?i thi Rad(I) =< ip(x), P >,
trang d6 ip(x), P nhuda quy uoc. Khi d6 t[ip(x)]mE I, pn c I. Do D[x] la vanh Noether nen I hUll h?n sinh
1=<

[ip(x)]m, pn, fr(x), f2(x), ..., it(x) >,

trang d6 ii(X) (1 ::; i ::; t) thuQc D[x].

19


Do I chCla trong Rud(I) =< :p(J'),P > nen Ill') E< ;(,r), P >, Suy fa
Ji(x) = :p(x)hi(:r)+ p;(l'),
trang do h;(x) E D[x], Pi(X) E P[x], V~y

1=< [p(x)]m, pn, ;;(x)h1(:r) + P1(X),,.., cp(x)ht(r)

+ Pt(x) >

,

({=) Truoc tien ta chung minh Ii- D[x]. Gia sl'i I = D[x]thl 1 E I va do do

1 = [,,(:r)!mu(x)+ llV(X)+
t

=

[cp(x)]rnu(x)+ ;;(x)

t

[\O(X)hi(X) + P,(X)]g:.r)

L h;(.r)gi(X)+ pu(:r) + L Pi(:r)g;(x)
k=l

i=1

t

=

(p E Po)

t

t

'P(x) (['P(x) ]m-Iu(x) + {; hi(x )9i(X)) + (PU(X)+ ~/i(X

)9i(X)).

D~t
t

r(x)

L h;(X)g,(l'),

= [:p(x)]m-1u(x)
+

k=l

t

s(x)

= pv(x) + LPi(X)gi(X)
;=1

E Pix:,

thl

1 = cp(x)r(x) + s(x),

Suy fa
1 + P[x] = :p(x)r(x) + P[x],

hay
1 + P[:r]= (y{r) + P[x])(r(x) + P[.r}
Tli ding thuc \'1:iathu dliQ'c ta suy fa :p(x) + P[x] kh?l nghich trang D[x]jP[x],
mall thu~n. V~y I i- D[x].
Do [:p(x)]mE I, pn c 1 nen theo Bo"dl 3.1.3, ta co I nguyen sa voi can toi cl~i

va Rad(1) =< cp(x),P >.
20


Hqn lilia, do ;(.r) Gon khdi lien ['P(x)]m-I,[hi(x) cho [:;(£)]"'-1 va Pi(X) cho [ip(x)]m,ta nh~n dliQ'Ccae phan du tliong. ling

Ia 1li(X)

Vat';(x).

V~y

1 =< [;-(£)]111.pT!, ;P(£)UI(X) + VI (J:), ..., ;:(£)lLt(.r) + L't(.c) >,

trang do u;(x), L'i(x)(l ::; i ::; t) thuQc D[x] thaa degui(x) ::; deg[degv;(x)

::; deg[.;(x)]m.

D(it hi(x) +- Ui(X) va Pi(X) +- Ci(X), ta

co di~u phai

chang minh.\?

3.1.5 Bif di. Cho D ld mien nguyen Dedekind, P ld ideal nguyen t6'khdc kh6ng
cua D. Khi do,
1) pn[x] = (P[x]t, Valm9i n nguyen du:ang.
2) C6 Po E P saD cho p[x] + pn[x] =< Po+ pn[x] > .
Chlcng minh. Kha don gi~ln lien dliQ'Cba qua.
Tv fJtnh Ii 3.1.4, ta c6 hai m~nh d~ sau

3.1.6M~nh dl. ChoI ld idealcua D[x]chua ;p(x) va gid szl Rad(I) =<

'f?(x), P

>.

Khi d6 I nguyen sa va 1=< Val n nguyen dl1ang.
Chung minh. Do Rad(I) =< 'f?(x), P > la ideal toi d~i (fJtnh /f 2.6) lien I
nguyen so. Do P C Rad(I) lien co n nguyen duong thoa pn C I va gia Slr

n la s6 nha nh?lt thoa di~u ki~n nay. D~ th?ly < c I. NgvQ'c l;,li,
ta se chang minh I C<;p(x), pn >, di~u nay tliong QUang voi chang minh

J c<

> trang D[x] = D[x]jpn[x].

Ta co Rad(J) =< y(x), p[x] >. VI D la mi~n nguyen Dedekind lien co Po E p
tho

a P[x] =<

Po >.

Voi f(x) E J, ta se chang minh bang qui n;,lp theo i (1 ::;i ::;n), f(x) bi~u di~n
duQ'c dvoi d;,lng
f(x)

=
trang do gi(:r).hi(x) E D[x].
21


Do f(x) E I nen

Tr0E RllIl(I) =<

, suy fa c6 91(X), III(x) E D[x] sao

cho f(x) = ';(1')gl(:r) + pl,hI(J;), V~y ph at bieu dung vaii = 1. Gia su phat bieu
aCing vai i

=k

(1 ::; k ::; 11 - 1), nghia la co g.,(x), hd:r) E D[x]saG cho

M

= f(X)gk(X) + p~hdx),

Suy fa p~hk(x) = f(x) - f(X)gk(X) E 7. Neu p~ E 7 thl pk[:r]C 7, d{lnden pk c I,
mall thu{'in \'6i k < n. \)y p~ ~ 7. Do I nguyen so nen hk(x) E Rad(l) =<
.;(x), Po >, suy fa c6 u(x). dx) E D[x] saG cho hdx)
f(x)

V~y f(x)

= ';(X)gk+l (x)

= .;{x)u(x) + pov(x).

=

rp(x)gdx) + p~hk(:r)

=


=

f(X)(~

+p~U[;)) +p~+IV(X)

+ p~+lhk+l (x) vai gk+l (x)

v(x). V~y phat bieu dung vaii

V~y

=k+

= gdx)

+ p~u(x) va hk+l(x)

=

1. Theo nguyen ly quy nq.p, vai mQi

= CP(~)gi(X)
+ pbhi(x), trong d6 gi(X), hi(x) E D[x]. Vai i = n,
ta co f(x) = .;(x)gn(x) + Pohn(x) = cp(x)gn(x) E< . Do d6 7 chua trong

i (1 ::; i ::; n), Hi)

< hay I C< . V~y I =< cp(x),pn >. c::;

Hoan roan tl1ang tv m~nh d~ tren, ta co

3.1.7 M~nh di. Cho I za ideal cua D[x]chua p va giGszl Rad(I) =< .
Khi do I nguyen s6 va 1=< [

P > v{ji m nguyen duong.

Nhl1v~y ta da m6 ta xong Calltruc cua ideal nguyen so trong D[x]. Tuy
nhien dq.ng cua ideal nguyen so trong Binh ii 3.1.4 chl1agQn, han nua rhea
Binh ii 1.5.6 mQi ideal nguyen so trong D[x] la giao clh hUll hq.n cac ideal
kh6ng the rut gQn c6 Cling can, nen tier rhea ta se 016 ra dq.ng clla ideal
kh6ng the rut gQn.

22


3.2 Cliu true eua cae ideal khong thi rzlt gQIltrollg

D[l:]

Trang nwc nay, ta se m6 ta c1c ideal kh6ng th~ rut gQn cua D[:!;].Do trang
r

,

vanh Noether I11Qi
ideal kh6ng th~ rut gQndell nguyen sa Hen l11Qiideal kh6ng
nguyen so dell co th~ rut gQn. VI the, trang nwc nay ta chI c:ln xet trang t?P
dc ideal nguyen so thay VIt~p tat ca cac ideal cua D[x].
Call truc cua cac ideal kh6ng th~ rut gQn vai can toi ti~u duQ'cm6 ta qua
dinh ly sau

3.2.1 Dinh Ii. Cho I ta ideal czla D[x]. Khi do, neu I nguyen sa vai din kh6ng
t6l dC;Zi
thi I kh6ng the' nit g9n.

Chung mink. Gia Sll I nguyen sa vai can kh6ng toi d?i co Rad(I) = J va I
co th~ rut gQn. Do I nguyen so nS'ican kh6ng toi d?i Hen rhea Dinh Ii 3.1.1
1= Y. Do I co th~ rut gQn Hen rhea M4nh di 1.4.8, ta co Ii, 12 nguyen sa,
I] =I I =I 12 thoa I = I] n

h va Rad(I1)= Rad(I2)= J. Theo Dinh Ii 3.1.1, co

1.:],k2 nguyen dliang sao cho Ii

= Jkl

va 12 = Jk2. Kh6ng mat tinh t6ng quat,

ta co th~ gia su k1 ~ 1.:2,khi do I] ~ 12, Sur ra I = h, mall thuan. Do do I
kh6ng th~ rut gQn.Q

Nhu v~y, rhea Dinh Ii 3.2.1, mQi ideal nguyen sa vai can kh6ng toi d?i
d~u kh6ng th~ flIt gQn. Tuy nhien, mQt ideal nguyen so vo1can toi d?i kh6ng
nhat thiet la ideal kh6ng th~ flit gQn; ch~ng h?n, xet D = Z va trang Z[x],
ideal < X2,pX,p2>=< X,p2 > n < x2,p > nguyen so vai can toi d?i va co th~
rut gQn. f)~ m6 ta Call truc db cac ideal kh6ng th~ rut gQn vai can toi d?i, ta
c:1n mQt so be) d~ sau

3.2.2 Bfl d€. Cho I ta ideal czla D[x], I =< [cp(xW,pS > vai T,s nguyen
dl1o'ng. Khi do T,s ta eae s6~nguyen dl1ang nho nh{[{ thoG [
23


Chung minh. Giii Sl( T kh6ng la so nguyen dl(cfngnho nhat thoa ['P(:r)j"E I,

nghla la co k nguyen dl(ang, k <
g(:r) E D[:r:], Ps E ps

T

sao cho ['P(xW E J. Do

do

t6n t~t'i f(x),

sao cho
[
sur fa
[+ P[x] = ['P(x)t f(x) + P[x].

t((c la
([
= «p(x) + P[x]Y(J(x) + P[x]),

nen
1 + P[x]= «p(x) + p[x])r-k(J(x) + P[.rJ,
hay
1 + P[x] = «p(x) + P[x]) [«p(x) + p[X]y-k-l(J(l')

+ P[x])].

Sur fa duang nho nhat sao cho [
Gia su s kh6ng la so nguyen duang nho nhat thO3 ps C I, t((c la co I
nguyen duong, I < s sao cho pi C I. Xet I trang D[x]= D[xJlPS[x]. Ta

co I =< [>. Theo Eo?dl 3.1.5, co Po E P thoa pi[x]=<

Pb

>. Vi

pi C I nen Pb = [
Rad(pI[x])) nen h(x) E pI[x],sur fa h(x) = Pbk(x). Do do Pb= P&[hay
Pb (1

- [
-

[
0 nguyen sa ,oa Rad(O)= F). Do do 1 + P[x] = (['P(xW + P[xJ) (k(X) + P[x])
hay [dt(ang nho nhat sao cho

ps

c

I. Q

3.2.3 Eo?dl. Cho I ld ideal nguyen SrJc6 din tOldr;1icua D[x] vai Rad(I) =<
. Gr;;im,' n ld ale 56 nguyen dZ1rJngnhd nh6.t saG cho [E I,
24


P"

c I. Khz'd6c6 k, I nguyen duong, k ::; II va I

::;rn, (hoa

1:< [ip(x)]m-l > = < ip(x), pk > .
I:pn-l[x]

=

< [ip(x)]l,p>.

Chung minh. Ta chung minh d~ng thCrcthu nhat. Neu Tn= 1 thl
1:< [ip(x)]rn-l> = I: R
=

I.

Mi;itkhac do [y(xW E I nen rhea M~nh di 3.1.6, ta co I =< ip(x),pr > v6'i r
ngllyen dLio'ng. Ho'n mia, rhea Bo?di 3.2.2,

T 1a

56 nguyen dlro'ng nho nhat

thoa p,. c I nen r = n. \)y b6 de dlfQ'cchung minh v6'i Tn= 1.
Neu Tn > 1, ta co [ip(x)]rn= [Do [Rad(I :< [1:< [=< ,

trang do k nguyen duo'ng.
Theo Bo?di 3.2.2, k 1a so nguyen dLio'ngnho nhat thoa pk c< . Mi;it
khac pn C I nen pn C I :< [=< ip(x), pk >. V~y k ::;n va d~ng thuc
thC£nhat dlfQ'Cchung minh xong.
Hoan roan tLio'ngt\j, ta co d~ng thC£cthC£hai, nghia 1a co I nguyen duo'ng,

I<

Tn

sac cho
I: pn-l[x] =< [ip(X)]I,
P > .v

3.2.4 Btf di. Cho I lil ideal czla D[x]{hoa [56/nguyen duang; han naG, 1:< [=< ip(x), ps > ho?fc I: ps-l[X] =<
[ip(x)]",P >. Khi do

1=< [ .
Chzlng minh.

Do [ip(xW E I, ps c I nen < [ip(x)Y, p5 >~ I. Ta se chC£ng

minh I ~< [ps >.
25


Xet tniang hQ'p [ :< [-;(xW-1 >=< <.p(X).
ps >. Trang D[x] = D[x]! PS[x], ta
se chCtng minh [ =< . Voi mQi f(x) E [, ta c6 T(X) E [ : cp(X)"-1=<

~

~

>. Suy fa t6n t~i

E

~

thoa

f(x)

=

CP(X)Ul(X). Gia Slt f(x)

=

cpk(X)Uk(X),
k < f, khi d6 .pk(x)udx) E [, do d6 udx) E [ : cp(x)k. Do k < l'
nen [: <.p(X)kC [: p(X)r-l.

= <.pk+l(X)Uk+l(X),

V?y f(x)

1 ::; i ::; T. Voi i

=T

Suy ra c6 Uk+l(X) E D[x] thoa

-

~

=

<.p(X)Uk+1(X),

Theo nguyen Iy qui n<;lp,f(x) = <.pi(X)Ui(X),
voi mQi

ta c6 f(x)

= cpr(x)ur(x) E<

-

<.pr(x)>. V?y ta da chung minh

du'Q'c I c< <.pr(x)
> hay [ c< <.pr(x),ps >.
Xet

tru'ang

hQ'p [ : ps-l [x]

=< [. Trang D[x] = D[x]1PS[x], ta

se chung minh [ =< <.p(x)>. Voi mQi f(x) E I, ta c6 f(x) E [ : PS-l[X] =<
-

-

~

,

[:p(xW,P[x] >=< [-;(xW.po >, trong d6 P[x] =< Po > "oi Po E P (do Bo de
3.1.5). Suy ra
f(x)

= [cp(xWhdx) + pok1(x)

Bang qui n<;lp theo i(l ::; i ::; B), ta chung minh du'Q'c
f(x) = [;;(xWhi(x)+ pbki(x)

Voi i = s ta du'Q'c
f(x)

=

[
=

[
V?y I c< hay I c< [ps >.
T6m I<;li,[ =< [:;(xW,ps > va b6 d~ du'Q'Cch{jng minh xong.\7

3.2.5 Bo?dl. V6'im9i n nguyen duCJng,ideal [=< kh6ng thl rut g9n
trang D[x].
Chung minh. Gia Slt [ =< <.p(x),
pn > co th~ rut gQntrang D[x],nghia la c6
Jr, [2 Ia c:k ideal cua D[x] sao cho
[

= Jrnh
26

(1)


v6'i It

1=

I

1=

12.

Ta c6 ';7(.£)E I, pn c I nen f(x) Ell, 12, pn ell,

h. Thea Btf di 3.1.3, ta c6

11.12ngllyen Sd va Rad(Id =< r;(x),P >= Rad(I2)' Thea M4nh di 3.1.5, t6n
t~li nl, n2 ngllyen dltdng saa cha
II

= < ,

12 =

(2)

< .

Ttt (1) va (2) ta c6

< =< ;(x), pnl > n < .
Da vai fro cua nl va n2 nhv nhall nen ta xem nl 2: n2. Khi d6 <
chua trang < <;(x),pn2 > nen
< =< ;(x), pnj > n <

.

V~y
< y(x), pnj >=< ,
hay

II = I,
mall thu~n v6'i di~u gia SUoT6m l<;li,I =< la ideal kh6ng th~ rut
gQn clla D[x].Q

3.2.6 B6?di. Val m91 m, n nguyen dl1(jng,ideal I =< [;(x)]m,pn > kh6ng thl
rUtg9n trang D[x].
Chling minh. Ne'u m = 1, rhea B6 di 3.2.5, ta c6 di~ll phai chung minh.
Xet trLtang hQ'pm > 1. Thea B6?di 3.2.2, m, n la cac so ngllyen dvdng
nho nhat thoa [.p(x)]mE I, pn C I. Thea B6?di 3.2.3, ta c6
1:< [cp(x)]m-I>=<

27

(3)


VCJih: :::;n. Tier theo, U se cht'tng minh h: = n, nghia I~ chang rninh 7 :<
[-;(X)]IIL-I >=< ;(x) > trang D[x]= D[x]/P"[x]. Dt?thay < \f(x) >c 7 :<
I

[~(x)]m-l >. :'\gu'Q'cl~ti,v6'i mQi f(x) E 7 :< [, ta c6 f(x)[
7 =<

[;p(x)]m >.

hay ['P(.r)]m-l (f(X)
f(x) -
Suy fa c6 g(x) E D[x] sao cho M[c;(x)]m-l

= [
~

- 'P(X19(X)) = O. M;;U khac [= Rad(O) nen
O. Do d6 f(x)
E< . V~y 7 :< [;(x)]m-l >=< .

V~y k = n va tu (3) ta c6
1:< [cp(x)]m-l>=< .

(4)

Bay gi6', ta cht'tng minh I kh6ng th~ rut gQn. Gia SlTngLtQ'cI~ti,tltc la c6 cae
ideal II. h db D[x] sao cho

1 = it

v6'i it

=1=

I

=1=

n 12

h. Suy fa

I :< [;(:r)]m-l >

n 12) :<

=

(II

=

(it:< [cp(X)]m-l » n (I2 :< [;P(X)]m-l ».

[
(5)

Tu (4) va (5) ta c6
(II :< [» n (h :< [

Theo Bli di 3.2.5, < kh6ng th~ rut gQn nen (it :< [
) =< cp(x),pn > ho?c (h :< [. Hdn nua [it,I2, pn c it. h nen rhea B6 di 3.2.4, ta c6 it =< [pn > ho?c 12=<
[pn >. tlTc la 11 = I ho?c 12 = I, mau thu:1n. V~y I =< [pn >
kh6ng th~ rut gQn trang D[x].Q

Tu B6?di 3.2.6, li~u c6 phai mQi ideal kh6ng th~ rut gQn c6 can la <
a~u c6 d<;1ng< [pn > hay kh6ng ? Cau tra l6'i la kh£ng ainh
va ta can mQtso b6 a~ sau

28


3.2.7 Rti dl. Xel vanh diGphllorzgRAIlvai R = D[x] va /vl =< cp(x),P >. Khi
d6 « [cp(X)Jk,pI »M

Chung minh. (~)

=

«

> hI nlu va chi nlu k = k' va l = L'.

[Ct/(x)]k', pt'

Hi~n nhien.

(==::;,)Gia si't « [cp(x)]k.pl »M

=

«

[cp(xW',p/ >h/'

Do \'ai tfO clb k va k' nhu nhau nen ta co th~ gia
, si't k ~ k'. Ta chung minh

k = k'. Th?t \'?y, gia si't k > k'. Ta co

[cp(~)]k

E «

[cp(X)]k',pt'

>hl

= «

[cp(x)]\ pi > hI nen co f(x), g(x) E D[x], PI E pi va m(x), n(x) E D[x] \ M sao
cho
[;(r)]k'

I

= [;(X}]k f(x) + PI
-- g(x)
1 m(x)
1 n(x)'

Suy fa

m(x)n(x)[;(x)t' = [f(x)]kf(x)n(x) + PIg(x)m(x),
nen
m(x)n(r) :;(X)]k' + P[x] = [cp(X)]k
f(x)n(x) + P[x].
Gian lu'Q'cn(x) + P[x]. [;(x)]k' + P[x] ( do n(x) + P[x], [;(x)]k' + P[x] =I-0 + P[x])

hai v~ clla dang thuc \-ua thu du'Q'c,ta co
,
m(x) + P[x] = [cp(x)]k-kf(x) + P[x],
suy fa m(x) E< y(x), P >= AI, mall thu1n. V?y k = k',
Do vai tfO clh l va [' nhlt nhau nen ta co th~ gia si't l ~ z'. Ta chung mini)
,
l = ['. Gia si't l > ['. D~t I =< [cp(x)]k,pi >, J' = [pi >, Khi do, trong
D[x] = D[x]/pl[rJ, ta co

Y=< [
l' =< [,

trong do < Po>= P[x] vai Po E P (do Rd dl 3.1.5),
,

-

-

--

VI I = I nen I = 1'. Do d6 I M = l' M

7

7

Ta co Po E< [;(x)]k >.\1 nen Po
1
1

=

[( )]k

1

29

~
m(x)

voi f(x)

E D[x]M' m(x) E

D[x]\ A{. Dodo co s(x) E D[.r]\ 1\1sao cho
P~m(x)s(x)

= [cp(X)]kf(x)s(x).(

*)

-I
SHY fa [cp(x)]kf(x)s(x) E< P~ >. M?t khac [cp(X)]krt p

,

f(x)s(x)

,-

E< p& >, SHY fa f(x)s(x)

dLtQ'C

,
p~

m(x)s(x)

-

-

-

= Rad(<

I

P~

» nen

= p~h(x), h(x) E D[.r]. Ket hQ'p v&i (*) ta

=

,[cp(X)]kp~h(x).

SHY fa

~ (m(x)s(x)

- [cp(x)]kh(X))

=

O.

M~itkhac P~ i=0 nen m(:r)s(x) - [;(x)]kh(x) E Rad(O) = P. SHYfa m(x)s(x) E<
[cp(X)]k. P

>c -'I. Mau thuc1ndo m(x)s(x) rt !vI. V~Yl = I'.r;)

3.2.8 Blf die Cho 1 la ideal ala D[x],1 =< [cp(x)t,ps > v6i t, s nguyen durJng.
Khi do
1: P[x] = < [cp(x)t, ps-l >.
1:< .p(x) > = < [cp(X)t-l, ps > .
HrJn mIa, voi AI la h? nhdn czla D[x], fa co

hI:

1M : (P[xDA! =

«

[cp(x)t, ps-l

> )M,

=

«

[cp(X)t-l,p$

»AI.

«

cp(x) »M

Chung minh. Trang vanh D[x] = D[x]jPS[x],ta co 1=< [cp(x)jr> . Bau tien
[a ch:"tng minh 1: P[x] =< [cp(xW, ps-1 > hay

I

:< Po >= < [cp(:;:)]r,
p~-l >, tfOng

d6 P[x] =< Po> v6'i PoE P (do Rlf di 3.1.5). Do PflJ50 =

Po

= 0 nen p~-l

E

I :< Po >. Hdn mla [cp(xWE I c I :< Po> nen < [cp(XW,pg-l>c I :< Po >.
NguQ'cl~i neu f(x) E I :< Po>, b~ng qui n~p theo i (1 :::;i :::;s - 1),ta se chung
minh
f(x) = [
30


--trang do h(x), J.;i(:r) E D[x].
VI f(1:) E

I

:< Po > nen pof(x) E I, do do pof(:r) = [:p(x)]rg(x). M~Hkhk

[cp(xW tt< Po> nen

~

E< Po> (do < Po> nguyen to ), do do g(x)

=

Puh(x).

hay
Suy fa pof(.c) = [.p(xWpoh(x)
Po(7(X) - [~(:~Wh(X)) = 0 (**).
Ma Po

=1=

0 nen f(x) - [-;(x)]rh(x)E Rad(O) =< Po >. Suy fa f(x) = [cp(x)]rh(x)+

P6g1(X), v~y (*) dung yO'i i
f(x)

=

=

1. Gia su (*) dung v6'i i = k < s - 1, nghia la

[cp(x)]rh(x) +P§9k\X). Ta co f(.r) - [cp(x)]rh(x)

= p~9dx),

ket hQ'p v6'i (**)

ta dltQ'c P~+19k(X) =

O. \1 k+ 1 < s nen p~+l =1=0, suy fa 9dx) E Rad(O) =< Po>,
hay 9dx) = PO9k+l(X),9k-l(X) E D[x]. V~y
f(.r) = [:p(x)]rh(x) + P~+19k+l(x)

Theo giil thiet qui n:;1p
f(x)

= [:p(x)]rh(x)+ pb9i(X),

v6'i mQi 1 :::;i :::;s - 1.
V6'i i = s - 1 ta duQ'c

f(x) = [.;(x)]rh(x) + pg-19s-1(X) E< [cp(X)]r,pg-l>
V~y ta da chung minh dltQ'c I: PIT] =< [cp(xW,ps-l >.
Tiep theo ta se chang minh I :< y(x) >=< [y?(X)r-l, ps >, hay chung minh
1:< y(x) >=< [:p(x)]r-l >.

D~ tha'y < [:p(x)]r-l,Fg>c I :<

~

>.

NguQ'c l<,liv6'i mQi 7[X) E

I

:<

cp(x) >, ta co f(x)y?(x) E I =< [y?(xW >, sur fa f(x)cp(x) = [y?(x)]rh(x),

= 0. Ma ",(x) rt Rad(O)= Plx[ nen
f(x) - [Y?(x)]r-lh(x)= 0 va do do f(x) E< [y?(x)]r-l>. Sur fa I :< cp(x)>c<

h(1') E D]Xj,

hay

:p(X) (1(1)

-1
[Y?(x)]r-l>. V~y ta eta chltng minh dltQ'c1:< y?(x)>=< [cp(xW-l,ps >.

31


Theo M~nh di 1.4.9),

(! : P[:r;])M = hf : (P[:r:]hf

va (I :< cp(x) »M

= hI

:«

y(:r) > Lu va do do ta thu dltQ'Chai d5ng tht'tc con l~li.Q

Hai b6 d~ sau duQ'ctrich

tu [9] (Dinh if 34 va Dinh

if 35 ctla chlldng 4).

3.2.9 Bil di. Cho R ld l)(lnhdia phuongNoethergiao hoan co don vi va

f

ld

ideal nguyen srJcua R co Rad(1) t6l dc;zi.Khi d6 neu f kh6ng the' rUtg9n thi
f: (f : J)

= J vai m9i ideal J chua f.

3.2.10 Bil di. Cho R ld vanh dia phuong Noether giao hoan co don vi va

f

ld

ideal kh6ng the' rUtg9n Clla R, J ld ideal cua R chua f. Khi d6 J kh6ng the' rUt

g9n neu va chi neu f : J chinh modulo f.
136d~ saul;) chi~u ngltQ'ccua Bo?di 3.2.6

3.2.11 Bo?di. Cho f ld ideal kh6ng the'rUt g9n trong D[x] vai Rad(I) =<
-p(:r),P >= JI. Khi d6, fon tfli m, n nguyen durJngsaGcho
f

=< [;p(x)]m,pn > .

Chung minh. D?t
S = { Q

IQ

kh6ng th~ nit gQn trang D[x], Rad(Q) =< tp(x),P >, Q i-<

}

[cp(x)]a,pb > voi mQi a, b nguyen dltdng
Ta se chang minh S = 0 b~ng phan chung.

Gia su S i- 0. VI. D[x] la v~mh

Noether nen S c6 ph~n ttt toi d?i la fa. VI. Rad(Io) =< y(x), P > nen c6 m, n
nguyen dvdng saD cho [,p(x)]m E fa, pn C fa va ta c6 th~ xem m, n la cae so
nguyen dltO'ng nho nh{{t thoa tinh chat nay.

Ne'u m = 1, rhea M~nh di 3.1.6, fa =< cp(x),pn >, mall thuKn VI fa thuQc
S.

32


l\i2\1

f/,

= 1, theo M41llz df 3.1.7, fo = [;(J,,)]TII,
P >, mf1LlthLl~n.

Nell Tn > 1 \':1 n > 1, rhea Blf df 3.2.3,

ta co

10 :< [~(x)Jm-l > = < ~(x), pk >,

(6)

fo :< pn-l[xJ > = < [ip(X)JL,
P >,

(7)

trong c16 1 ::; k ::; n va 1 ::;l ::;117.
Neu k = n ha?c l = m, rhea Bo?df 3.2.4, 10=< [~(x)Jm.pn >, mall thuan.

Gia stf k < n va l < m.
D?t S' = { hI
);\1. /."

=1=«

1

hI kh6ng th~ n.'1tgQn trang (D[x])Ju, Rad(I1\1) = «

[ipCr W. pb

ip(:r), P >

> ~,'J voi mQi a. b nguyen cllfo'ng }.

Ta co (Io)Jula ph:ln tv toi d~i ci.'taS'. Th~t v~y, ghi Slfco (It)M thuQc S' saG cha
(Iohl chua trang (IdA/. TnJoc lien, ta chUng minh II thuQC S. VI (IdM kh6ng
th~ flit gQn nen It kh6ng th~ rut gQn va II ~ M. Sur fa RadIt ~ AI; m;)t khac,
Rad((It)M) = (Rad(It)).u = « ~(x), p »M va Rad(Id nguyen to nen rhea Binh

/f 1.4.10, ta co Rad(I1J=< . Ho'n nua II

=1=< [~(x)Ja,pb

> vCJimQi

a, b nguyen dVa'ng,VIneu co ao,bonguyen dvo'ng thoa It =< [ip(x)Jao,
pbo> thl
(It)M

= «

[r.p(x)]ao,pbo

> L\1, mall thuan. V~y It thuQc S.

Tiep thea, ta chung mint 10 ~ II. Gia sv 10 :l It; ta co u E 10 \ It. Sur fa
'!.:E (IO)M; va da do '!.:E (IdA! nen co 9 E It,h rt !v! saG cha '!.:= -hg. Sur fa
1
1
1
uh = 9 E II. VI It kh6ng th~ rut gQn, da do nguyen so', va h rt AI = Rad(1d
nen u E It, mall thuan. V~y 10~ It.
VI 10 la ph:ln tlf t6i d~i cua S nen 10= It. Sur fa (IO)M= (Idlvl. V~y (10)1>1
la
ph:ln tlf t6i d;;1icua S'.
Tiep thea, ta se chung minh co T,s, U,v la c:k 56 nguyen dvo'ng saG cha
(f0: P[X])lvI

(Io :< ip(x) »M

=

«

[~(xW,pS

»A/,

= « [~(x)Ju,pv > hI.

(8)

(9)

TrVCJclien, da m, n la ck 56 nguyen dvo'ng nho nhat thoa [~(x)]m E 10,pn c 10

33


V~l 11/. I/.

> 1 nen
(10)M ~

(10: P[x]),\!.

(/O)M

~ (fo:< :p(x) »M,

(1ohJ

(10)

(/0 : P[.T])AJ,

=I

(/O)M =I (/0:<

;p(:£)»M.

(11)

Do (fa)M kh6ng th2 rut g()n nen rhea Dinh Ii 3.2.9
(/0);\1 : [(/oh! : (10 + P[X])M]

= (/0 +

P[x])A!

hay
(foh!

: : fo).\! : (fa -+-P[X])M]

= (fO)M + (P[x]h!

M<;itkhclc do P;.r] la ideal nhan nen (P[X])iV!chinh, tu day rhea Dinh Ii 3.2.10,
(fO).\1 : (fo

+ P[l'])M

kh6ng

th~ [(it gQn .

M<;itkhac
(fOi.\! : (fo:-

P[l'DAf

=

(10)M: [(/oh!

+ (P[X])Al]

= [(10)M: (1ohrJ n [(fohl
=

: (P[X])M]

(10)1\11:(P[X])M

nen (fohI : (P[.r])M kh6ng th~ nIt gQn.

Do Rnd(foh! = «

:p(x).

P »M nen Rad((10)AI: (P[X]hI) =

«

y(x).

P > )M. V?y

(fo : P[:T])Afkh6ng th~ rLItgQn trong (D[X])M,Rad(1o : P[x])J',1= « :p(x), P > hI.
Ho'n nCta, (10).\[, ph~n tlTtoi d?i ct'Ia 5', chua trong va khac (fa : P[X])AInen
t6n t?i T,S nguyendtTong

saG cho

(10 : P[xDA[ = «

[rp(x)t,

ps

> )Af.

V?y ta co (8).
Hoan roan tuong t\T, ta thu duQ'c (9).
B~ng quy n?p rhea i(l :::;i :::;n - 1), ta se chung minh
(10: pi[X])M

=«

[tp(x)t,ps-i+l

34

»Af.


Voi 'i = 1,

tu

i = j(1

:::; 11

:::;

j

(8), ta co di~l1 phai chung minh.

Gi!l Sli ding thlic Cling voi

- 2), nghia la
(10 : pJ [J;])M =

«

[:p(x)]'",

ps-j+L

»Al.

Suy fa

= (ps+L[X])MC (10)M

(ps-j+1 [X])M(Pj[X])M

do do pHI C 10. Soy fa 8 + 1 ;:: n.

Do 1 :::;j :::;n - 2 nen (8 + 1) - j

=8-

j + 1 ;:: 2.

Ta co
(10: ph-I[X])Al

=

[(10: pj[x]) : P[x]Lu

=

(10: pJ[.r]Lu : (P[x])Al

=

«

=

« [:p(xW, ps-j »M

V?y d:1ng thuc dung ,"oi i

[<;(~.W, ps-j+l

»M

: (P[X])M

Ctheo B(f di 3.2.8 ).

= j + 1. Theo nguyen Iy quy n?p, d:1ng thuc

(10 : Pi[xDA! = ~< [i

=n

- 1, ta co
(10 : pn-l [xDA!

=

«

[tp(xW, ps-n+2

> )M.

(12)

Tu'o'ng tv, tlt \9) va bang quy n?p ta co
(10 :< [.;(x)]m-l > ).\f =

«

[tp(x)t-m+2,

PV

>)M.

(13)

Tu (6) va (7) fa co
(10:pn-l[X])M
(10 :< [tp(x)]m-l »M

= «[hI,

(14)

=

(15)

35

«

Tli (12), (14) va Blf di 3.2.7 ta co
T

= 1..'3- n + 2 = 1,

hay
T=l,s=n-l.

(16)

Tu'o'ngtIj, tv (13) va (15), ta co
u - m + 2 = 1,v = k,
hay
u = m - 1,v = k.

(17)

Tv (8), (9), (16). (17) ta co
(fo : P[X])M
(10 :< ;(x) »M

=

«

> )M,

(18)

=

« [tp(x)]m-r, pk > )M.

(19)

[tp(X)]l,

pn-l

Tv (18),(19) va BII di 3.2.8, ta co
(10: P[X])M: « tp(x) »M
(fo :< tp(X) > hI : (P[X])M

=

«

[tp(x)]l-l,pn-l

=

«

[tp(X)]m-l,

»M,

pk-l

> )M.

(20)
(21)

Ho'n nU'a

(10 : P[X]);\I : «

hI

=

(10: P[X] < tp(X)»M

=

(10:< P[X])M

=

(10:< tp(X)»M : (P[X])Mo

(22)

Tti (20), (21) va (22) ta co
«

[tp(x)]l-l.pn-l

».\1

=«
36

[

»,'v/o

(23)


Tv (23) va Bd dl 3.2.7, ta co I - 1 = m - 1, n - 1 = Ie - 1 tlie la I = m, n = k,
mall thufin voi tru'O'ng hQ'p dang xet.
V;;ly, trang mQi tnlO'ng hQ'p, ta dell co mall thufin.

Do do S = 0, nghia la,

neLl I la ideal kh6ng th~ [(It gQn clla D[x] voi Rad(I) =< y?(x), P > th1,tcSnt<;li
m, n nguyen dvo'ng sao cho
I =< [~(.r)]m, pn > ,

C/

Tv Bif dl 3.2.6,Bo?dl 3.2.11"J Bo?dl 3.2.2,ta co

3.2.12 Dink Ii (M6
D[:r] fa khangthi

ra cac ideal kh6ng

th~ rut gQn voi can toi d<;li).Ideal I Clla

nit g9n vaidin tal dr;zinlu va chi nlu ton tr;ziideal nguyen ta/

PolO Clla D, da thuc don khrJi -p(.r)E"D[x] thod D[xJlP[x]va k, 1 nguyen during sao cho

1=< [,

trong d6 m, n nguyen during,

Hon naa, khz d6 Rad(I) =<

thoa [

> va m, n fa cac sa/nguyen during nho nhdt

pn c I,

Nhu' V?y, ta da m6 ta dIu true clla cae ideal kh6ng th~ [(it gQn trang vanh
da th{TCD[x], trang do D la mi~n nguyen.Dedekind. Tv s1j m6 ta nay, ideal

nguyen so' vCiican toi ti~u chinh li ideal kh6ng th~ flit gQn vai can toi ti~u.
Tuy nhien, ideal nguyen so' voi can toi d<;likh6ng nha't thiet la ideal kh6ng

th~ rut gQn vo-ican toi d<;li.Dung Dink Ii 1.5.6 va Dink Ii 3.2.12 ta co mQt
s1jm6 ra khac voi Dinh Ii 3.1.4 v~ d<;lngcih ideal nguyen so voi can toi d<;li
nhlT saD

3.2.13 H~ qua. Ideal I cua D[x]fa nguyen sri vai can tal dr;zinlu va chi nlu
37