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Factorial domains not satisfying the krullintersection propertyPaolo Zanardo aa Dipaimeto di Maematica , Pura e applicaa , Via Belzoni 7, Padova, 35131, ItalyPublished online: 27 Jun 2007.

To cite this article: Paolo Zanardo (1999) Factorial domains not satisfying the krull intersection property,Communications in Algebra, 27:4, 1717-1728, DOI: 10.1080/00927879908826524

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COMMUNICATIONS IN ALGEBRA, 27(4), 1717-1728 (1999)

FACTORIAL DOMAINS NOT SATISFYING

THE KRULL INTERSECTION PROPERTY

Paolo Zanardo

Dipartiinento di hlatematica Pura e Applicata

l'ia Belzoni 7 33131 Padova, Italy

INTRODUCTIOK

The famous Iirull intersection theorem shows that , if I is a proper ideal of a noetherian integral domain R. then (-)go I" {O}. 1% shall say that a (proper) ideal I of a commutative ring R satisfies the *bKrull intersectioi~ propelty" (slioitly KIP) if I' = {O} Nore generally, we shall say that R satisfies the Krull intersection property if every (proper) ideal of R satisfies the KIP.

\Then dealing with a factorial donlain R, it 15 clear that its piincipal ideals satisfy tlle IiIP. On the other hand. it is rather expectable that there exist non-noetherian factorial doinains which do not satisfy the IiIP. However. esamples do cot seem to be cornn~only known. Actually. tlle standard es- anlples of non-noether~an factorial domains do satlsfy the KIP: in fact, D.D. Anderson in [A] has pioved that , if D is a noetherian domaul and {Sx} is an? set of indete~minates, then D[{2yx}] satisfies the IiIP.

An esample. due to P. Eakin, of a factorial domain T not satisfying the Iirull intersection property may be found in the paper hy Houston. Lilcas anti L-iswanatha11 [HLL-] (Esample 5.7). We remark that thr authors of that paper just needed that Eakin's exaniple T mas a L u l l d o i ~ ~ a m (which was imnlediate by construction). Anyway. T turns out to be factorial, as one can easlly check nlaking use of Lemma 1.1 of this paper.

1717

Copyr~ght 0 1999 by Marcel Dekker, Inc.

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1718 ZANARDO

AIore recently. Yascimento in [XI construct,ed a factorial (lomain of tli- rnrnsion < 4 not satisfying the IiIP. Z'ote that Eakin's exanl1)le has infinite dimension.

The pluspose of the present paper is to provide a class of cxalnples of facto- rial tlolnains not satisfying the Iirull intersect,ion property. hIore specifically. in the, first scction we construct. for all r , > 2. a factorial domain R,, of dinlelision 7 1 + 2 n-hich contaills a two-generated ideal J not satisf>-ing the I iIP ( Theorc~n 1.2). I11 the second section we construct a factorial domain R silch that tllc finitely generated R-ideals satisfy the IiIP. but tllerc &st ~lol~-finitrlly generated R-ideals not satisfying the property ( Tlieore~n 2.1) . 11% selnarl; that a Bey point to s l m ~ that the dorilains R,, and R a s ahow are factorial is l)rovided hy tllc pas>- Lemma 1.1.

In 110th sectiorw. to p~rforni the c ~ ~ i s t ~ r u c t i o ~ i ~ of factorial dolnitii~s, we coml)illc the idea of Ealiill's exanlple i r ~ [HLT-] and the ideas developt~l 1,y D.F. Antlerson and IIulay in jL4SI]. Thc purpose of the paper [A111 n-as to constrllct non-noetherian factorial domains A,, (of dimension rz + 2) . not sati?f!-ing another most important thtorem by Icrull. namely the yen era l i z ed pr inc2po l idea l t h e o r e m . Several arglunents in the wrifications of the p r o p erties of the R,, are adaptations of those in [XLI]. Howe~-er. the factorial domains A4,, in [.All] sat,isfy the KIP. It is n~orth noting that Nascimento in [X] also l~orrowed tecllnicjues from Anderson and hIulap: as a collseqlut.ncc, tllc tliscussiol~ in [?;I 11ears some rcsenlblance to that of the first s e c t i ~ ) ~ ~ of the present papcr.

The autlmr is indehtetl to R. Gilmer and D.F. Anderson for having 5ug- grsted examination of the papers [AhI] and [HLT']. and to P. Eakin for useful comnlent s and sugg~st ions.

fll. TTc start n-it11 a lemma XT-hich d l 11e cruclal in proving unique factorva tion. 1,oth in tllii wction and in the next one.

Lemma 1.1. Let R be a dornaln, A, L? m~~l t iphca t~ r -e l~ - closed subset< of' R. and let T = Ra n RB If I;, E A is a prune element of RB. the i~ p is alio a pnmc t*lemeilt of' T

P r o o f Let us suppo5e that f g E pT, fol irutable f. g E T . Since T C R E and 11 is prime in Ra. vie may aisume. without loss, that f = p h , ~ i t h 11 E Rs Yaw j E T c RRa. ~qhence a150 h = f / p E RA. and therefore 11 E Ra n RE = T. L1-e conclude that p 1s prime in T

It 15 ~lnportant to remark that L e m m 1.1. together w t h Sagata ' i the orenl. can be ;tl)plled to s h o ~ that the example bg Eakin, quoted 111 the introduction, of a Iilull tionlam not satisfying t h ~ KIP. is actuallv a factor la1 do~naln (see Exalilple 5.7 of [HLT*]).

Let 11s now state the main lebult of the present section

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KRULL INTERSECTION PROPERTY 1719

Theorem 1.2. For everj- n > 2 there exists a factorial domain R, of dimen- sion 7~ + 3. containing two-generated ideals which do not satisfy the I < d l intersection property

The remainder of the section ill he devoted to the construction of such R,, and to the verification of its propertlei. As already mentioned in the intloduction. the definition of R, is inspired by that of A, in [AhI]. adding the itlea of Eakin's example in [HLI-1. The proofs of the following claim and of step E ale adapted from [iZhI].

Let S. Z. I;, I;', 1 < I < rx be indeteiminates over the field I<. Fol 1 < 2 < x let 3, = CJzo b13-YJ Zl he elenlents of IC[[X, Z]]. n-hich are algeb~a~cally independent over the field I<(-\-. Z )

FOI all 1 < 2.3 < m w e s e t

It is then iinlnediate to see that , for all 2 . 3 , we haye

For a fixed 72 > 2 we denote by s and t those unique integers > 1 sllcll that: 12 = a + t ; 0 5 t - .5 < 1 (so that .s = t when I I is eye11 and t = s + 1 ~vllen 7i is odd). This notation will be standing for the remainder of the first section.

Now we define the domain

As a consequence of the relations ( I ) , we see that R, = UZ1 Rn,. where

Note that the R,,, are factorial domains. sillre X. Z. the ITT3 and the I ; , ale algclj~aically independent when j is fixed.

Let us considel the two tlomains:

Let u s note that the definitions of the and of the I;, imply that

and. symmetrically.

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1720 ZANARDO i

R,[Z-'1 = Ii[S. Z. Z - I , I;'. . . . .I;'. LT1,, . . . , IT,, : 1 5 1 < x].

11-e shall use tllc fact that R,,[Z-'1 = U:, R,,,[Z-'1, whelc each R,,3[Z-1] is a UFD. A4rlaloqouily. Rn[S-'1 = UFl R,,,[S-'1, and each Rn3[S- ' ] i i a UFD.

In what follows we &all apply sexera1 times Kagata'i theorem on factol~al tlomains. fol xvliich we rrfer. for instance, to [I<]. Theor. 177, 11. 131

CLAIM. R , , [ A y ~ l ] and R,,[Zp'] are factonal domains, and Z a11d S a i r prime elc~rlerlt i of R,, [-Y-'1 a i d R,, [Z-'1 , respectir-el>-

Proof. K e follow the arguments by =Inderson a d Mulay. 1Te 4lall proxe that S is a prinle elenlent of the factollal donlam R,, [Z-'1, the proof 15

wnllal for Z and R,, [I-'].

i ) S is a prinle element:

in fact. let f y E XRn[Z-'1. with f . y E R,[Z-l] Since Rn[Zp'] = Upl R,,,[Z-'1, we may mite fy = S 7 , ~vllere f , y . h E R,,[Z-'1. for a

J lalge enoue;h Smce -Y 11s prime in R,,[Z-'1. we may assume that f E S R , , J [Z-'1 c -YR, [Zp'] , from which the aqseltlon

ii) (-)El S' RT1[Z-'] = (0) :

let us set S = Ii[[S Z]][I'it. . . . . I;']: we will embed R,,[Z-'1 into S[Z-'I. ITe define a nlorphisln 4 : Ii[-y. Z, k; , . . . . I , . I.;'. . . . , Y;'] + S through the assignrnerlts S H aY7i, Z H Z. I; F- - ; S t , for 1 < i < s, and I;I H I;!. for 1 5 I . < t . Thcn Q is injective, since the 3, are algebraically independent o\-er I i (S Z ) Hence 6 niaj lje estended to an nyective honlonxxph~iln hetn-ern the le5pect1ve fields of quot~ents Here,

is factorial. Thils R,,[Z-'1 is a Iirull domain, being the intersection of two

factwial tlomains. alld turns out to be factorial, by Nagata's theorem. since S is prime and R, [Z-'. S-'1 is factorial.

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KRULL INTERSECTION PROPERTY

TTP now p~oceed b>- steps.

A. R,, = R,[S-'1 n R,,[Z-l]

Proof. It suffices to show that any f E R, [-y-l] n R,[Z-'1 lies in R,,. Let us choosc j large enough to have f E RnJ [-y7i-1] n RnJ[Z- ' ] . TVe may write f = f i / 3 = f 2 / Z k , for suitable f l , f 2 E RnJ and IT. k E N . The equality Z V l = Sh f z : valid in the factorial domain R,xj. implies that S h divides f l in RnJ . since -Y and Z are tion-associate prime eleme~it,s of R n j . Tlien f E R,,, c R,. as desired.

In view of the preced~ng claim. R, is a Iirull cloma~n, being the interiection of two factorla1 domains.

B. R, is a factorial domain. and A, Z are prime elements of R,.

Proof UP shall a p p l ~ Nagata's the or en^. Let S bc the multiplicative set generated hy S and Z. TVe have to show that (R , ) s is factol~al, and that Ay m t l Z ale prune elr~nents of R,, . Obserx-e that

whence (R , , is a UFD. The fact that X , Z ale prime elementi of R, i i a consequence of the above clalm. of step A and of Lemma 1.1. By Kagata.5 throrem, we conclutl(- that R,, 1s factorial. as desired.

Let us no\+ coniide~ the ideal J = (S. Z)R,, of R,. Let us note that .Y/Z Sf R , , [ S 1 ] and Z/-Y S f R,,[Z-'1 implies that -Y and Z are lion-associate p~ imcs of R,, .

C . J 1s a mamnlal ideal of R, Afore preclselj. ewsj elerrlent of R,, is erther In J or congruent to a constant of I< modulo J

Proof Since the last statement is an immediate consequence of the relations ( 1 ) . it suffices to sllow that J is a proper ideal By contradiction, let us iuppose that

1 = A - f + Z y with f , g E R,.

Let us fix j large enough to ensure that f. g E Rn3. Then R,, = (S Z)R,,, . which IS clearlv impossible, since R,,, is a polynomial ling in the indetermi- n a t e ~ -7i. 2. I',,, 15 2 15 r st .

so that R,, does not satisfv the I h l l intersection property

Proof. First of all, let US note that

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ZANARDO

for cx-er;- I 5 5 . From the relations (1) we clerive. for all k > 1.

from (2 ) we mlrnecliately reach the deiired conclusion.

For an! domaln R with field of fractions L , we denote by dim, R t h e u n l u a t z r l ~ dz7rzrnszon of R (see [GI. Ch. 30 p. 360). Recall that

dirn,.R = sup {dim D : R C: D C: L ) .

E. tiirrl R,, = dim, R,, = 11 + 2. Proof A5 alxxe, let S be the multiplicative set g e ~ l e ~ a t e d by S and Z. Let ui r ecall that

I < [ S Z . I ; , . . ,I;, k;'. . . . .I;'] c R,,

C I < [ - Y . ~ ~ - ~ . Z . Z - ' . I ) . . . . . IYq. I:,. . . .I;] = ( R , , ) 5 ,

Sow. a i a conwyumce of Th. 30.9. p. 360 of [GI. xve have

rhence dim, R,, = 11 + 2, too. I l o r e o w ~ dim, R, > tlinl R,, > dill1 ( R n ) 5 = ii + 2. n-l~enct- d m R,, = 12 + 2. as deinetl

REMARK. Comparing our R, with the factorial donlains -4,. constructed 17;- Anderson anti LIulay in [illl]. one co111d asli if the <A,, do satisfy the IiIP. The answer is affirmative. Howewr. we have preferred to omit the proof of this fact, since it is not crucial for the present paper. and ~ ~ o u l d need new definitions anti rather long verificatiuns.

42. The wxmd section is devoted to proving the following

Theorem 2.1. T l l ~ r e exists a factonal domam R, not satlsfvmg the Iirull ~rlteriectlon propert>, such that the p r o p a t y 15 satisfied by ~ t s finltek gell- cratcd idcali

111 the sul~sequcnt arguments. we shall need the not,ion of power series ring in illfinite indeterminates. We refer to Gilmer's paper [GI] for definit,ions and main results on this niatter. In [GI] it is pointed out that there are three different ways of defining power series in infinite indeterminates. We s l d l

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KRULL INTERSECTION PROPERTY 1723

use the third and most gene~al definition of poweI 3erles rlng. denoted 111

[Gl] by D [ [ { - y ~ ) ] ] 3 ( D is a commutative ring and { X x ) is an Infinite set of indeternlinates).

lye shall need the lmportant fact that I<[[{Sx)]13 is factorial when I i is a field: this is a corollary of more general results by Cashwell and Ewiet t [CE] .

Let I., I r ' , S , , Z,, 1 5 r < x, be indeternlinates oxer the field I<. Let us choose 6, E I i in such a way that

is transcendent over the field I<(S, , 2, : 1 < r < x). Let us set ITI = Iw/X1. 1; = 17'/Z1 and, for j > 2,

To simplify the notation me shall write

From the ah(>\-? definitions we readily get the following relation5 for j > 1:

Our purpose is to show that R is a factorial domain which satisfiei the propelties ~eyuired in the statement of Theorem 2.1. In several places the argument.; will follow the same lines as in the first section However. since the notation 13 a little awkward, Tve tllought it convenient to gire full detalls

Let 115 note that the lelations ( 3 ) imply that R = Uzo D L . ~vheie

is factorial. since X,. 2,. [ T I . T i , 1 < r < k . are algebraically indepeilclent over I<.

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1724 ZANARDO

Let .I' and 2 br the multiplicatirely closed subsets of R generated by tllp ST. and by the Z, . respectively (1 < I . < cc). 5% use the l~otation .I' = (-Y, r < X) aud Z = (2, : r < x) 5% also set Xk = ( S 1 . . . . ,-YTCI) mcl Zn = (ZI , . . . . Zb).

Let us first olxerl-e that

In fact, let f E R,u n R z . lye want to show that. necessarily, f E R. By the tlcfinition of R. tllerc exists a k 2 1 large enough such that f can he w i t ten in the f o ~ i n

Thus n-e ohtain the folloviing relation, d i d in the polynomial ring DL :

Since the ZI are prime clernents of Dk. pallwise non associate. ~t follows that n,<, -Yri clirides f l in D L . whence f E DL C R

Kow we note that. as a consequence of ( 3 ) . we have

is factorial. since it is a localization of a polynonlial ring (X,. 2,. 1: S i . 1 < r 5 X - , are algebraically independent over I<, mllen k is fixed). ilnalogously

is factorial. lye sllall piore T-arious properties of R.Y, n-hich will imply that R,u is

factorial: ailalogous arguments show that RZ is factorial, too.

Proof. Let 11s suppose that ub = Z, f . with a. h. f E Rx . TVe can choose k such that Z,, n. h. f E -AL; since -4 is factorial, we may assume that a = Z,g with g E .-Ik c Rs. from which the assertion. 13

Let 11s set L = IC(I.*) and let us denote by C the power series ring Y = L [ [ S , . Z, : 1 < r < C C ] ] ~ .

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KRULL INTERSECTION PROPERTY 1725

2. Tliere exists an embedding 9 : R x + C x such that, for all r . q : -I-, H S,, 9 : Z,. H Z,, p : I' H Y , and p : I-' H 3.

Proof. lye first define a morphism 3 : I i [X , . 2, , I , I-'] + S through the assign~nentb X , H -Xr, Z , H Z,, I - H I- and I" H 3. Then 9 is injective, since 3 is t~ansceilclent over I i ( X , , 2 , ) . Hence q inay be extended to an mjective homomorphism between the respective fields of quotients. Herr.

Tllui p ( R , y ) C Y,u . as desired.

3. 0, Z:Rs = (0) for every r E N

Proof In fact 0, Z:Ys = { 0 } , smce Y x I? factorla1 and the 2,. are not umts of Y u . The assertion follows fiom the existence of the eml~eddlng ,-

Let u i now set -4 = 0, (R ,y ) (Z> and B = JR,Y)_7 It is plain that B = I i [ S , , 2,. I: 17'],y 2 is a factorial domain.

P ~ o o f . Each (R ,u ) ( z , , 15 a DVR. by virtue of 1 and 3. We hal-e to verify that ewrp a E -4 1s invertible in alnmst all the DT7Rs j R x ) ( z T ) . Since n = b/c here b. c E R y , it lr enough tc~ show that b 1s divisible in R x at most by a finite number of Z,, and t h ~ s happens, since p j b ) is C~T-isible in the UFD Y s at no st by a finite ilunlber of Z ,

5. R s = -4 n B. and therefbre R s is a Ifiull domain.

Proof. l l e 1ia.r-e to show that R x > -4 n B. Let y E A n B Since i/ E B. n-r may TI-rite y = c / d , where c E R x and d E 2. lye need to show that d di.i ides c in Rev It clearly suffices to prove that any prune factor of d. say Z,, necessarily divides c. In fact, x ,: -4 implies that x = a / b with a , b E R,y and h @ (2 , ) . From ncl = bc we obtain that 2, div~des c , iince Z , is a prime

elenlent of R s by 1, it dlvitles ad and does not divide b

6 . R x is a factorial domain

Proof. Tllr tlomaln = I i [ X , , Z,, I.: I-'],y,z is factorial. and 2 is gen- erated by p~ in le elements of R x . Since R,l is a Iirull domain, then R,y i i factorial. hy ~ ~ r t u r of Kagata's theorem.

Let us lemark that R 2 is also factorial. and satisfies properties simlar to those of RAY. as one can check with the appropriate, obvious changes.

7. R = R x n R 2 1s a factorial dornam and the elen~ents S,. and Z , are all pnme 111 R

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1726 ZANARDO

Proof. R is a Iirull doniairi, being the intersection of t w ) factorial tlonlains. Each S, is primc in Rz and each Z , is prime in R x : therefore, as a conse- cl~ience of the Lr~lima 1.1, the ele~nents S, and Z,. of R are all prime in R. l loreowr R,y,_7 = I<[S, . Z,. 1; I.-' : 1 < r < , ~ ] , y , ~ is a factorial donlain. By Sagata's theormi, R is a factorial donlain. too.

It is worth noting t,hat the prime elements S,, 2,. 1 < 1, < x are pairn-ise 11011 a.ssociate in R. In fact. S,./Z, carillot lie in any --Ik;, so it does not lie in R,y > R: similarly, X,/_Yj cannot lie in any Bk if i # . j . and so on.

Let ui no\v consider the ideal *\I = (S,. 2, 1 < 7- < x ) of R. T I > h a l l sllow that I f is a lnasilrlal Ideal of R. and that nk &\Ik # (0) thelefolc R tloei not qatiify the IiIP

Proof Since <k. E MA. it is enough to prow the following relation. for all k 2 1 :

In fact. from the relationr ( 3 ) i t follaws

T k l m e now to prow that .\I is a proper arid maximal ideal. This fact. together wit11 the result that R sat irfici the KIP for finltely generated ideal\, nlll 1)e prowd t l l r~ugh a i e~ ies of steps.

For each X 2 1. we consider the ideal -Ifk = (S,. Z , : r > k ) . In thi.; notat~on. = -lJ1

9. Let f he an element of R. if X > 1 1s sllch that f E D L . thc311

Proof Since f IS a polynomial 111 S 1 . . . SL. Z1, , Z k , I- , , 1;. \Te read& get oul asiertion from the relations ( 3 ) .

10. Let f t R. for. all pos i t l~e integer s there exists a polj~noimal q in the lndetrmmateq .Til,. . . .A-3, Z1. . . , Zs S I I C ~ that f = q (Llf3+l)

Proof. Let us choose 1; 2 5 such that f E D k . From 9 we der~ve the es- istence of a pol! nomial p in S , . . . . X k , Z1.. . , Z k , congruent to f modulo IZfk-+l Since X 2 5 . we have Jfn+l C 3I,+1. and we immediately reach oul conclusion.

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KRULL INTERSECTION PROPERTY 1727

11. Let 0 # p be a pol,-nornial in the indeterminates -TI, . . , ,Yk, 2 , . . . , Z k . regarded as an elernent of R: then p $ l\lk+l.

Proof. B y contradiction. we assume that there esi5t f l . . , fn .g l . . . , g n E R such that

Let us pick s 2 X. + 12 such that all the f , , y , lie in D,. Regarding to ( 5 ) a5 a relation valid in the polynomial ling D,. we get p = 0, a cuntr acliction.

12. -21 is a maximal ideal of R.

Proof. From 11 we see that the nonzero constants of Ii are not elements of J f = -U1, so that -L1 is a proper ideal. From 10 we see that every elenlent of R is congruent to a constant of Ii modulo -11. and therefore -21 is a nlasinlal deal.

U P can say much more: actually we have

13. For all positive integer k we have R/Afk+l Ii[-Yl,. . . . -Yx. Z1.. . . Zx] Proof. B y 10 every f E R is congruent, moclulo dlk+1. to a polynomial p E Ii[X1,. . . . X r . Z1 . . . . . Zr]. By 11, given p E I i[_Yl. . . . , -Til,. 21. . . . . 2x1. we have 11 E 0 (?.Il+l) if and only if p = 0. These facts readily ~ ~ r l d our statement. 0

14. Let I = ( f l . . . . f h ) be a propel fin~telp generated ideal of R, then

n,, = (01 Proof. B y contiadiction, let U? a5wrne that 0 # f E On I n . Let u s choo5e X > 1 such that f . f l . . f h E D L Kote that, as a conwqucnce of (3) . a11 tlle element s I - , . I j lie in tlle domain

where S = ( S T . 2, : k + 1 5 T < x). Since f E n, I n . for every h-tuple ( ~ a l . . . , x h ) E ~ h s ~ ~ c h t h a t n l + . . . + n h =n , the reex i s t a(,, R.sucll that

Lye may regard the a(,, .,) as elements of T. Thus the a(,,, ,,,) are rational f~mctions whose deno~ninators are products of X , and 2, with 7 > X. + 1. The specialization = 1 = Z,, for every r > k + 1, turns (6) into a relation of the f o ~ m

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where the g(,,, ,,,,, , E D k . It follows that 0 # f E ni,(fl.. . . . f h ) n D k . ~ h i c h is a cont,radiction. s i ~ m ( f l . . . . . f h ) D k is a proper ideal of the polynomial ring D k .

[A] D. D. .\nclerson. T h e Krull an,tersect~on the ore^^^. Pacific J . hlath. 57 (19751, 11 14. [ . I l l ] D. F Anderion and S.B. hlula>-. Non-catenary fa,ctorzal donmzlss. C'omm. in .klge-

bra 17(5) (1989). 1179-1185.

[CE] E.D. Cashwell and C..J. Everett. Formal power serzes, Pacific J . hlath. 13 (1963). 4 5 - 6 .

[G] R . Gi ln~cr . Multzplzcatzt~e Ideal Theory , hfarcel Uekker. Tie\%- York. 197%.

[GI] R. Gilimer. Po,wer s m e s rsngs over a K r d l domatn . Pacific J . . l lath. 29 ( l ! ) f j U ) ,

54:j-.5 19, [ H I S ] E. (;. Houiton, T .G. Lucas a i d T.11. \ ' isnanathan. Prt~rsery decon~posrtrons of

dzvlsorlal d e a l s rn Morz dorna~ns , J Algebra 117 (1988). 327-312. [li] I I<aplal~sl.;y. C~ommtr ta t~tre Rzngs. The ITniversity of Chicago Press. Chicago ailcl

Loi~cfon, 1.97-L.

[ U ] h l . C:. Nascimento. Intersectzor~ o f powers of przme ideals t n Kr.ul1 d o r n n z ~ ~ s . C o m i ~ l . 111 -4lgehra 20(3) (1992). 777-782.

Received: September 1997

Revised: March 1998

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