C A U T R U C C U A I D E A N N G U Y E N T i C t l A V A N H » A TB&C Pb^mThlBlchHi*
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Bdi bdo gidi thiiu vi cdu trie cua idian nguyen ti bit Id trtmg vdnh da thic mit biin va idean nguyin to don thuc tiong vdnh da thic nhiiu biiit
Tit khda: Vdnh, idetm nguyin ti, vdnh da thic.
1. DAT VAN Dfi
Cho R l i vinh giao hota cd don vi. Mpt iddan / thpc sp cfa /i 14 idean nguyOn t i neuvdimpi a.b^R v4 abf,I suyra a e / hota A e / . Vipc ttm Ufe tinh chit v4 ciu tiric cfa idean nguyen t i cfa mO. vanh cho tmde dupc nUfe ngndi quan tim ngUto cto. Cic vfe d l niy dupc trinh biy trong [1], [2], [3], [4] v4 [5].
Mpc dich chinh cfa b4i bta fay l i hp fliing 1 ji v i tiinh biy cU tilt cta chtog minh cho vipc md th ciu tiiic cfa id6an nguyen t i ttong vanh da ttnic. Cta kit qfa fay dupe tttoh biy ttong [2], [3], [4] v4 [5] dndi djng chti f v4 bii tjp.
Ngoii phta gidi tUto, bii bta cUa tiiinh fai mpc. Muc 2 md t i ciu ttiic cfa id6an nguyto td bit ki ttong vinh da ttnic mOt biiu (Dinh I^ 2.2). Muc 3 md t i ciu ttiic cfa idean nguyto td don ttita ttong vinh fa ttnic nUlu bife (Djnh If 3.4).
2. VANH DA THtrc M O T B I S N
Trong mpc fay, chfag ta ludn gii ttriet JC [x] l i vSnh da ttnic bifa x ttto tiutog K.
Ttndc hit, a nhic Iji mOt ke. qfa quen bilt sau:
Mpihe^2.1.
Vinh / : [x] Ii v i t a cta idean chinh, ngUa l i mpi iddan dfe suih bdi mdt da flita.
Dpa vio mpnh d l nen ta cd thl chiing minh dnpc kit qui chinh cfa mpc niy nhn sau:
Binh ly 2.2.
Gii su / la idean cfa v i t a K.\x\. KU do. / li iddan nguydn t i khi v4 cU kU / = ( « W ) , ttong dd y(ji:) 14 da ttnic bit kh4 quy hofc da ttidc 0.
' Gidng vien khoa Khoa hoc Ttr nhien, Truang Bai hoc HSng Btic 78
T j J C r t KBOA H P C TRtroING BAI HOC H 6 N G B t r c - s 6 29.2011!
ChiiBg mjnh;
•'=>" N f e / = 0 kU dd til c d / = (0). N f e / # 0 , ttieo Menh d l 2.1 til cd ttil vilt / = ( « W ) , ttong dd j ( x ) * 0 . Giasu ? ( j ) khdng bit khi quy, ngUa la tdn tji hai da thuc q,(x) vi q,{x) sao cho ?(x) = , , ( x ) . , , ( x ) , d c g 9 , ( x ) < d e g j ( x ) v i d e g ? , ( x ) < d e g ? ( x ) . Gii s u ? , ( x ) e / k U d d tacd ttil vilt , , ( x ) = A(x).g(x), ttong dd A(x)eiS:[x]. Suyra deg?, (x)> deg j ( x ) , man ttiuta vdi d e g ? , ( x ) < d e g ? ( x ) do do 9, (x) 0 / . Chtog mtah tirong tu ta ctag cd q, (x) g / . Nhn viy q, (x).q, (x) e / m i
?, (x) « / v i J, (x) g / nto / khdng phai Ii idcan nguydn td. Dife fay miu ttiuta vdi gia ttrilt / l i idean nguydn td. Vjy ? ( i ) Ii da ttito bit khi quy.
"<="Nto / = (0) flri/nguyto t i . Nfe idean / * 0 . ttieo Mpnh d l 2.1 til cd ttil vilt I = (q(x)) vdi q(x)*0 vi q(x) bit khi quy. Gii sri f.fsl, ttong dd f.-AsK[x]. Tfe tji / 3 ( x ) e J f [ x ] ttifa mta y ; ( x ) . / , ( x ) = / , ( x ) . , ( x ) . Vi q{x) bit k h i q u y n t o t a c d f,(xy: q{x) hota A(x) I ,(x).NgU-al4 f,(x)^I hota / , ( x ) e / hay /14 Iddan nguyen td.
Nto K14 ttutog s i phto ttii ta cd kit qfa quen ttiupc sau:
Bi ai 2.3. Gii Sli C[jc] 14 vinh fa ttiirc ttto ttutog s i phto C . Khi dd fa ttito / ( x ) kbicfattitoOcfa C[x] bit khi k U v i c U k U cddang / ( x ) = <ix+4,ttong dd a*0.
H p * ™ 2 . * Gii s i r / l i idcan cfa vinh C [ i r ] . K U d d , / l i Idean nguyto td khi v i cUkU / = 0 hota / = ( a r + * ) . ttong do a * 0 .
Chihig minh:
Theo Djnh Iy 2.2 v i Bd dl 2.3 tii nhta dupc dife phii chtog mtah.
3. VANH DA THirc N BIEN
Trong muc niy, chtag ta luta xe. K[x,....,x,] Ii vanh da ttita „ b i f e ( « > l ) t t t a tiudng K. D l cbo gpn ta vii. ^ h . . . x . ] = / : [ x ] . Cta k l , qfa ttong mpc niy dupc tttah biy ttong [2]. [3] vi [5].
Vinh fattlto „ bife vdi „ > 2 khdng phii 14 viUl cbririi. Do viy viec md . i cfe
^ c u a Idean nguyto td ttong vtah fa ttito nUfe bife Ii khdng don gita nhu v i t a md.
bito. Trong phjm vi bii bao fay chtag .di cU xem xe. ddi vdi Idp idcan don ttito '
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De tife cta v i giii quyi. dupc vfe d l niy, tmde hit cbiii^ tM xrai xdt t t t o vilril mpt bife. Ta ttiiy rfag idean don thfc tixmg vinh mdtbifeicd djng / = ( x " ) . Theo Djnh Iy 2.2 ta cd / nguyta t i Wri v i cU kU i " bit khi quy. Dd dd a = l , n ^ a l i I = {x).
Chtog toi gidi flripu chung minh sau tay dii hon so vdi cich diiing minh ttta nfaung giiip chtog ta tife cta dupe bii tofa tiong trutog fapp nUfe biln. Dd l i nOi dung chinh cfa dtah ly sau:
Binh if 3.1. Gii su / # 0 14 iddan don flnic cfa v4nh ^ [ x ] . / l i idean nguyto l i k U v i c U k U / = ( x ) .
Chihig minh:
" = > " G i i s u / = ( x ) . Liy tiiy ^ da flifc / = i 2 „ + a | X + . . . + a , / ' e / . ICUdd tacd fill vilt / = g ( x ) j r , tiongdd g = * j + i ^ x + . . . + 6 „ i " . T a n h t a d u p c :
a„-l-atX-i-...-l-a,x"=(b^-i-bjX-t-...-l-b„x')x
Dfeg nhit ttifc hai v i t a dupc <%=0. Do dd / ( x ) 6 / k U v i c U k U / ( x ) cddjng:
f = ajX-t-...-^a„x"
Liy hai da thfc bit ky:
/ ( x ) = < i , + a , x + . . . + a , x " . g W = 6 o + i , ^ + . . . + A . x - e ^ [ x ] sao cho / ( x ) . g ( x ) e / .Ta cd:
/ W - g W = a „ i „ + ( a | 6 j + a „ J j ) x + . . . i - J a / t x ' + . . . + ( a , f e . ) x " "
j+k=i
Vi / ( x ) . g ( x ) E / . n e n 0^=0.00 Oj,*, e X , ndn ta nhta dnpc oj, = 0 hojc 6„=0, ngUali / ( x ) 6 / hojc g ( x ) e / . V i y / I i Idean nguydn ti.
" = > " Gii s u / = (x'"), ttong dd a > 1. ICU dd x ' = x.x°-'
Tacd x' Bl nbung x e / v i x " ' « / . Dta dfe miu ttiuta vdi / nguyto t i . Vjy a = I b a y / = ( x ) .
Biy gid ta xet ttto vinh da ttifc nUlu bife ( K > 2 ). Trudc hit m cfe mOt s i kit qfa b i ttp. Khdng mit tinh tdng qfat, til cd till gii si: flni tp cta bife 14 x, ,x ,...,x
KiUeu: Z,_ =(x,^^ x J vdi 0 < A < « - 1 . Trong dd X,=(x„...,x,) haytacd
Bi di 3.2. Cho f{X) 14 da ttifc ttto vinh A ' [ X ] . KU dd / ( J " ) ludn vilt dnpc dudi djng sau:
I » f c m KHOA H q c . TRlfONG BAI HQC HpNG Bt'C - S 6 29.2(111
/ m = a Wx, +....+g.(jr,^)x, +ft„(A'.)
trong do: l<k<n vi gj(Xj_,) e K[Xj_,] v6il<jsk + l.Ctaihigniiiih:
G i i s i r / ( J - ) = 2 ; a , . f ' '
Budc I: Nhdm at ci cta tu cfa /(jr) chfa bife X, ta dupc:
/ W = g , ( J O x , + / ( ^ , ) (,) Budc 2: Nhdm tit ci cta tii cfa / (x„) chfa bife X,^ m dnpc:
/;(j^,)=gi(^,)x,+/,(x,_) PJ Tiep nic lim nhn vjy dfe AKOC i ta dnpc:
/.-.('f,„) = gi(Jr,,)x,, +ft.,(jr„) (^
COng (l)...(k)ta dnpc:
/ W = gi W x , +...+g.(jr,_Jx,_ +g,.,(A-4) Vjy bd dl dupc chfag murii xong.
Bidi3.3.Chof{X)sK[X]vi / = (x, x,.). D a t h f c / ( ; r ) e / k U vicU kU f{X) viit dupc dudi djng nhn sau:
/ W = g,(Jr)x, +....-l-g,(.X,Jx,^ vol l<k<n ttong dd gj(.X,JsK[X,^J vdi i<j<k.
Chihig minh:
"=>"liytayydattluc/(jr)e/Theo Bidi 3.2 tacd / W = g,(.y)x, +....+g.(Jr^^)x,, -^g,.,(^,)
^ - y ^ ft««.) = / W - g , W x , - g . ( A ' , ^ ) e / '
Theo[2,Bdde4.2viBddl4.3L«cdctatircfadaftfc g..,(x,) phaicUabl.
cho mo. ttong efc bife x,^ vdi (y = U - I ) . Vifattlfc g,,,(X,J khdng ehfa cta bife
•"t i^i.ntocictucfa g,„(X,J phii btagO, dodo g,„(X,J = 0.
Vjy ta nhta dupe /=g,(jr)x, +....+g,(A',_ )x^ .
• c - Gii su / = g,(X)x, +....+g^(X.)x,, ttico dtoh ngUa cfa id6an smh bdi tjp { x ^ , . . . , x j t a c d / ( j r ) s / .
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Vta dpng cta Bd de trto ta chi^iiiinh dupe.k^ t ^ d^di cda bii bta.nhn:sau:
^ l A (^ 5.* Gii sur/la idean don ttnic tiin vinh, C[X]. KUidd if 14 idSta l^uj!^ ii td kU vi cU kU / stoh bdi tjp cta biln, ngba li / ={xj.,,..,x ),. tiong,dd. ISk^n- •
Chirng minh:
"<=" Gii sri: / = (x^,...,Xj^) vdi l < * < n . Khdngmit tinh ttag quit, tacd&tgii sii ttni tn cta bife Ii Xj,...,x^.liyhaidattriic g(X) vi «C^"tiSyffln^c :ciX].Thta'i Bd dl 3.2, ta cd thl viit:
g{Xhgii^^,,-^--+g„(X,Jx,-tg,JXJ h(X)^h,(X)x,^ +....-,-h,(,XJx,_ +h^(X,-) ttongdd: g,(X,^).h,(X,JeK[X,^lliJik-i-l. Bit:
a W ^ , + +gi,(^iJh = « , vi h,(X)x^ + +Aj(4^)x^ = a . KUdd: g{X)=B^-t-g,JX,^) vi h{X)=B,+hUX^) Suyra:
g(X)J,(X)=[B^ + g,,,(X,J\.[B,+h,^,(X,J\ I
= B,£,+B^K,,(X^)+B,g,„(X,;)+gUX„)K,,(X^) \
Dfe dfe:
ft«(^4)Ai.,(^^) = /(X)g(J')-fi^.B.-B/,^,(;r^)-.B,g,,,(X,.) Giisii g(X).A{X)6/. KUdd gUXOKt(.X„)^I.
TheoBddl3.3,tacd: gUXO-K,M,^) = a \ Vi K[XJ 14 miln nguyto, nto gi„,,(Xi^)aO hota Aiti(^,) = 0, nghia IE
g , ( X ) 6 / hojc Aj(Jr)6/.Vjy/14Ideannguytoto, !
"=>" Gi4sii /=(.4), ttongdd A Ii he snrii iii tilu cfa /
rn(*ig/i(ip/:Nfetontei xf e/i vdi a > l .Tacd J ' , nfaung x.jcf"'e /
Vtf ' '
mau ttiufe vdi ti'nfa nguyto ti cfa / .
Trudng hap 2: Gii su surii A cU chfa cta bife x, ,...,x^ vi A chfa don ttifc djng «
^k'-^i?' ('S«)-DoAIihpsuriitiitifentotacd Xj^" «.< vi X;'*.,j'^e.i. V|>/ ! khdng phiili idean nguyen ti. Hay .4 pbii cddjng 14 {x,_,...,X4} vdi ( I S i S n ) .
Vjy dinh ly dupc chtog nrinh xong.
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•E»rertKHOAH9fiTRtfaNGBAI Hgc H6NG Btic- S6 29.2016
[1]
TAi H p U T H A M K H A O
M. F. Atiyah m d l O, Maokmikf (1969), Intioduction lo Commutative Algebra, Addison-Wesley.
[2] J. Heizog and T. Hibi (2011), Monomial ideals. Springer Press.
[3] Le TnfeHoa (2013), Dgi si may tlnh, Nxb. Dji Hpc Qudc Gia.
[4] H. Malsumura (1986), Commutotive Ring Theory, Cambridge Umveisity Press.
[5] R.Y. Sharp (1998), Steps in Commutirtive Algebra. Cambridge Umversity Press.
T H E PREME ffiEAL S I H U C T U R E O F P O L Y N O M U L R I N G S
Pham Thi Bich Ha ABSTRACT
This paper intioduces arbitiary prime ideals stiuctiire of polynomial rings in an indeterminate and prime monomial ideals stiuctiire of polynomial rings in the indetermirmtes.
Keywords: iiing. prime ideal, polynomial ring.