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Formally Integrally Closed Domains and the Rings R((X))

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Formally Integrally Closed Domains and the Rings R((X)) JOURNAL OF ALGEBRA 200, 347]362Ž. 1998 ARTICLE NO. JA977262 Formally Integrally Closed Domains and the Rings RXŽŽ .. and RXÄÄ 44 D. D. AndersonU Department of Mathematics, The Uni¨ersity of Iowa, Iowa City, Iowa 52242 and B. G. Kang², ³ Department of Mathematics, Pohang Institute of Science and Technology, Pohang, 790-784 Korea Communicated by Craig Huneke Received June 20, 1997 Let R be an integral domain. For f g R@ X # let A f be the ideal of R generated by the coefficients of f. We define R to be formally integrally closed m Ž.ŽAfg tsAA f g . t for all nonzero f, g g R@ X #. Examples of formally integrally closed domains include locally finite intersections of one-dimensional PruferÈ do- mainsŽ. e.g., Krull domains and one-dimensional PruferÈ domains . We study the rings RXŽŽ .. R@X# and RXÄÄ 44 R@X# where N Ä f R@ X #< A R4 s NNs ts g fs and Ntfs Äf g R@ X #<Ž.A ts R 4. We show that R is a Krull domainŽ resp., Dedekind domain.m RX ÄÄ 44 Žresp., RX ŽŽ ... is a Krull domainŽ resp., Dedekind domain.m RX ÄÄ 44 Žresp., RX ŽŽ ... is a Euclidean domain m everyŽ. principal ideal of RXÄÄ 44 Žresp., RX ŽŽ ... is extended from R m R is formally integrally closed and every prime ideal of RXÄÄ 44 Žresp., RX ŽŽ ... is extended from R. Q 1998 Academic Press Key Words: formally integrally closed; Krull domain; PID; Euclidean domain; power series ring. U E-mail address: [email protected]. ² E-mail address: [email protected]. ³ B. G. Kang was supported by KOSEF Research Grant 951-0107-038-2. 347 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. 348 ANDERSON AND KANG 1. INTRODUCTION Let R be an integral domain with quotient field K. For f g RXwxor R@ X #, let A f be the ideal of R generated by the coefficients of f. The rings RXŽ.sÄfrgf<,ggRXwx,AgsR4Ä4Äand RX s frgf<,ggRXwx, Ž. 4 Ag ¨ sRhave been widely studied. For example, R is a PruferÈ domain mRXŽ.is a PruferÈ domainŽ in fact, a Bezout domain.m every Ž.principal ideal of RX Ž.is extended from R m R is integrally closed and every prime ideal of RXŽ.is extended from R. In addition R is a PruferÈ ¨-multiplication domainŽ.Ž PVMD i.e., every finitely generated ideal of R is t-invertible. m RXÄ4is a PVMDŽ. in fact, a Bezout domain meveryŽ. principal ideal of RXÄ4is extended from R m R is integrally closed and every prime ideal of RXÄ4is extended from R. Moreover, R is a Krull domain m RXÄ4is a PID. Also, recall that R is integrally closed Ž. Ž . m Afg ¨ sAAfg¨for all nonzero f, g g RXwx. These results along with necessary terminology and notationŽ especially concerning the t- operation. are reviewed in Section 2. The purpose of this paper is to consider the power series analog of the results mentioned in the previous paragraph. We define R to be formally integrally closed if Ž.ŽA fg ts AA f g . t for all nonzero f, g g R@ X #.If Ris formally integrally closed, then R is completely integrally closed, but not conversely. Examples of formally integrally closed domains include Krull domains, or more generally, integral domains that are locally finite inter- sections of one-dimensional PruferÈ domains. The power series analogs of the rings RXŽ.and RXÄ4are RXŽŽ .. sR@X#N where N s Ä f g R@X#< A R4ÄÄ44and RX R@X# where N Äf R@ X #<Ž.A R4. fNs s tts g fts In Section 3 we investigate formally integrally closed domains and diviso- rial ideals and t-ideals in R@ X #T where T is a multiplicatively closed subset of Nt. We show that if R is formally integrally closed and T : Nt , ŽŽ . Ž . Ž . Ž . then IR@ X # ¨ T s I¨ @ X # TTs I @ X # ¨s IR@ X #T¨for any nonzero fractional ideal I of R. We also show that if R is formally integrally closed and J is a finite type ¨-ideal of R@ X # with J l R / 0, then J s I @ X # for some ¨-ideal I of R. Because the results for RXÄ4involve every nonzero finitely generated ideal of R being t-invertibleŽ. i.e., R is a PVMD , we should expect the corresponding results for RXÄÄ 44 to involve every nonzero countably gener- ated ideal of R being t-invertibleŽ. i.e., R is a Krull domain . This is indeed the case. In Section 4, the results for RXÄ 4 are extended to RXÄÄ 44.We show that R is a Krull domain m RXÄÄ 44 is a Krull domain m RXÄÄ 44 is a Euclidean domain m everyŽ. principal ideal of RXÄÄ 44 is extended from RmRis formally integrally closed and every prime Ž.t-ideal is extended from R. We also show that R is a Dedekind domain m RXŽŽ .. is a Dedekind domain m RXŽŽ ..is a Euclidean domain m every Ž principal . FORMALLY INTEGRALLY CLOSED DOMAINS 349 ideal of RXŽŽ .. is extended from R m R is formally integrally closed and every prime Ž.t-ideal is extended from R. 2. THE t-OPERATION AND POLYNOMIAL RINGS U Let R be an integral domain with quotient field K. Let R s R y Ä40, MaxŽ.R be the set of maximal ideals of R, FR Ž.be the set of nonzero fractional ideals of R, and let FRU Ž.be the finitely generated members of Ž. Ž. Žy1.y1 Ä < 4Ä< FR. For I g FR,I¨s I sFRx I : Rx : K and Its D JJ¨ UŽ.4 Ž. :Iwith J g FR. Then I is called a ¨-ideal resp., t-ideal if I s I¨ Ž. UŽ. resp., I s It . Note that for I g FR,I¨ sIt. Every proper t-ideal is contained in a maximal t-ideal and maximal t-ideals are prime. We will denote the set of maximal t-ideals by t-MaxŽ.R . Here ¨ and t are examples of star operations. For an introduction to star operations and the ¨-operation seewx 13, Sections 32 and 34 . For results on the t-operation and t-invertibilityŽ. defined in the following text seewx 5, 14, 16, and 17 . Recall y1 that I g FRŽ.is t-in¨ertible if ŽII .t s R or equivalently, I has finite type U Žthat is, Itts J for some J g FRŽ..and IMis principal for each M g t- MaxŽ.R . An integral domain R is a PruferÈ ¨-multiplication domainŽ. PVMD U if every I g FRŽ.is t-invertible. For results about PVMDs, seew 14, 17, 19x . It is well knownŽ e.g.,wx 18, Theorem 3.6. that R is a Krull domain if U and only if each I g FRŽ.is t-invertible. Also, for R a Krull domain, Ž. It sI¨ for each I g FR and hence we will use these two star operations interchangeably. For f g KXwx, let A f denote the fractional ideal of R generated by the coefficients of f. It is well known that R is integrally closed if and only if Ž. Ž . UŽ U. Afg ¨ sAAfg¨for all f, g g RXwx or equivalently, f, g g KXwx. Because the ideals involved are finitely generated, we could of course replace ¨ by t. The implication Ž.« is due to Krull and may be found in wx13, Proposition 34.8 . Although the implication Ž.¥ is often attributed to Querrewx 20 , it was actually proved by H. Flanders wx 11 30 years earlier. Ä < 4Ä<Ž. 4 The sets N s f g RXwx Af sR and N¨ s f g RXwx Af ¨ sR are saturated multiplicatively closed sets. This may easily be proved directly, or Ä < Ž.4 seen by noting that N s RXwxyDMRwx X M g Max R and N¨ s RXwxyDÄMRwx X< M g t-MaxŽ.R 4. The two quotient rings of RXwx, RXŽ. RXwxand RXÄ4 RXwx, have been extensively studiedŽ for s NNs ¨ example, seewx 1, 2, 3, 6, 15, 17. In the next sections, we will study the power series analog of these two rings. With this in mind, we state some important known results about these rings. Although we state these results for a single indeterminate, they remain true for any nonempty set of indeterminates. 350 ANDERSON AND KANG THEOREM 2.1. Let R be an integral domain. Then PicŽŽRX ..s U PicŽRXÄ4.s0. For f g RXwx, Aisinf ¨ertibleŽ. resp., t-in¨ertible if and only if fRŽ. X s Aff R Ž.Ž X resp., fRÄ4 X s ARX Ä4.. Proof. The fact that PicŽŽRX ..s0 is given inwx 2 and the fact that PicŽ RXÄ4.s0 is given inwx 17 . The second statement may also be found in the respective sources. THEOREM 2.2. Let R be an integral domain and let T be a multiplicati¨ely Ž. closed subset of N¨ . Let I g FR.Then y1 y1 Ž.1 ŽIXwxTT. sŽIXwx., Ž. Ž . 2 IXwxT ¨¨sIXwxT,and Ž.3 ŽIXwxTt.sIX twx T. Proof. This iswx 17, Proposition 2.2 . THEOREM 2.3. For an integral domain R the following are equi¨alent. Ž.1 R is PruferÈ . Ž.2 R Ž X . is PruferÈ . Ž.3 R Ž X . is Bezout. Ž.4 E¨ery Ž principal .
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