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. Kang†, ‡
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 Prufer¨ do- mainsŽ. e.g., Krull domains and one-dimensional Prufer¨ 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. ᮊ 1998 Academic Press Key Words: formally integrally closed; Krull domain; PID; Euclidean domain; power series ring.
U E-mail address: [email protected]. † E-mail address: [email protected]. ‡ B. G. Kang was supported by KOSEF Research Grant 951-0107-038-2.
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0021-8693r98 $25.00 Copyright ᮊ 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 Prufer¨ domain mRXŽ.is a Prufer¨ 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 Prufer¨ ¨-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 Prufer¨ 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 Prufer¨ ¨-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 Prufer¨ . Ž.2 R Ž X . is Prufer¨ . Ž.3 R Ž X . is Bezout. Ž.4 E¨ery Ž principal . ideal of R Ž X . is extended from R. Ž.5 The map : L ŽR .ª L ŽRX Ž .. gi¨en by ŽA .s AR Ž X . is a multiplication preser¨ing lattice isomorphism from the lattice of ideals of R to the lattice of ideals of RŽ. X . Ž.6 R is integrally closed and e¨ery prime ideal of R Ž X . is extended from R. Proof. The equivalence ofŽ.Ž. 1 ᎐ 3 and Ž. 6 is given inwx 6 . The equiva- lence ofŽ.Ž. 1 , 4 , and Ž. 5 is given inwx 1 . THEOREM 2.4. For an integral domain R, the following are equi¨alent. Ž.1 R is a Dedekind domain. Ž.2 R Ž X . is a Dedekind domain. Ž.3 R Ž X . is a PID. Ž.4 R Ž X . is a Euclidean domain. Proof. The equivalence ofŽ.Ž. 1 ᎐ 3 is given inwx 6 . That the equivalence ofŽ. 1 and Ž. 4 follows fromwxwx 10, Theorem 5.3 is remarked in 1 . THEOREM 2.5. For an integral domain R, the following are equi¨alent. Ž.1 R is a PVMD, i.e., e¨ery nonzero finitely generated ideal of R is t-in¨ertible. Ž.2 RÄ4 X is a PVMD. Ž.3 RÄ4 X is Bezout. FORMALLY INTEGRALLY CLOSED DOMAINS 351
Ž.4 E¨ery Ž principal . ideal of RÄ4 X is extended from R.
Ž.5 The map : Lt ŽR .ª L ŽRXÄ4.Ž. gi¨en by A s ARÄ4 X is a multiplication preser¨ing lattice isomorphism from the lattice of t-ideals of R to the lattice of ideals of RÄ4 X . Ž.6 R is integrally closed and e¨ery prime ideal of RÄ4 X is extended from R. Proof. This is given inwx 17, Theorems 3.7, 3.14, and Corollary 3.16 . THEOREM 2.6. For an integral domain R, the following are equi¨alent. Ž.1 R is a Krull domain. Ž.2 RÄ4 X is a Krull domain. Ž.3 RÄ4 X is a PID. Ž.4 RÄ4 X is a Euclidean domain. Proof. Ž.1 « Ž.3 .wx 12 or wx 17 . ClearlyŽ. 4 « Ž.3 « Ž.Ž.2. 2 « Ž.1. This follows because RXÄ4lKsR.3Ž.« Ž.4 . Clearly RXÄ4has 1 in its stable range. Bywx 10, Theorem 5.3 , RXÄ4is Euclidean.
3. FORMALLY INTEGRALLY CLOSED DOMAINS
Let R be an integral domain with quotient field K. Therefore, as remarked in Section 2, R is integrally closed if and only if Ž.ŽA fg ts AA f g . t U for all f, g g RXwx. We define R to be formally integrally closed if U Ž.ŽAfg tsAA f g . t for all f, g g R@ X # .Ž Here, as in the polynomial case, Af is the ideal of R generated by the coefficients of f.. Inwx 4, Theorem 2.1 Ž. we showed that R is completely integrally closed if and only if A fg ¨ s Ž. UU AAfg¨for all f g RXwxand g g R@ X # . Thus if R is formally inte- grally closed, R is completely integrally closed, but the converse is false Ž.see the following text . We showedwx 4, Corollary 2.6 that if R is a U one-dimensional Prufer¨ domain, then A fgs AA f g for all f, g g R@ X # . So a one-dimensional Prufer¨ domain is formally integrally closed. We also showedwx 4, Corollary 2.7 that if R is a locally finite intersection of rank-one valuation domains, then R is formally integrally closed. Thus a Krull domainŽ. e.g., an integrally closed Noetherian domain is formally integrally closed. More generally, we have the following result.
PROPOSITION 3.1. Let R s F R␣ be a locally finite intersection where each R␣ is formally integrally closed. Then R is formally integrally closed. Proof. The proof is almost identical to the proof ofwx 4, Corollary 2.7 . In Proposition 3.1 we really do need the intersection to be locally finite as the following examplewx 4, Example 2.10 shows. 352 ANDERSON AND KANG
EXAMPLE 3.2. Let E be the ring of entire functions. Then E is a completely integrally closed Bezout domain that is an intersection of Ž. Ž . U rank-one DVRs. Hence A fg ¨ s AAfg¨for all f, g g E@ X # . How- Ž.Ž. U ever, we do not even have AŽ Xqb. gts AA Xqbgtfor all b g E and U ggE@X#.Ž Note that because E is a Bezout domain every ideal of E is a U t-ideal and hence Ž.ŽA fg ts AA f g . tmA fgsAA f g for f, g g E@ X # .. Thus while E is an intersection of rank-one DVRsŽ and hence an intersec- tion of formally integrally closed overrings. , E is not formally integrally U closed. Also, note that whereas Ž.ŽA fg ts AA f g . t for all f, g g R@ X # Ž. Ž . U implies that A fg ¨ s AAfg¨for all f, g g R@ X # , the converse is false.
PROPOSITION 3.3. Let R be an integral domain such that RM is a one-dimensional ¨aluation domain for each M g t-MaxŽ.ŽRi.e., Risa PVMD with t-dim R s 1..Then R is formally integrally closed.
U Proof. Let f, g g R@ X # . Now for M g t-MaxŽ.R , RM is a one-dimen- sional valuation domain and hence ARfg MsAAR f g M byw 4, Theorem Ž. 2.4xw . Then by 17, Theorem 3.5 x , A fg ts F Mgt-MaxŽ R. ARfg M s Ž. FMgt-MaxŽ R. AARfgMs AA fgt.
Let I be an ideal of R. While for the polynomial ring RXwx,IR wx X s IXwx, the same is not true for the power series ring R@ X #. Here I @ X # s Ä fgR@X#< Af:I4Äwhereas IR@ X # s if11qиии qifinn< jgI,f jgR@X#4 sÄfgR@X# THEOREM 3.4. Let R be an integral domain and let T be a multiplicati¨ely closed subset of R@ X #. Let I be a nonzero fractional ideal of R. y1 y1 y1 y1 Ž.1 ŽIR@ X # .ŽTTs I @ X #.Žs I @ X # .Ž TT: I @ X # .. If R is formally integrally closed and T : Nt , the last containment is an equality. Ž.2 If R is formally integrally closed and T : Nt , then I @ X # IR@ X # I @ X # I @ X # IR@ X # . Ž.Ž.Ž.¨¨TTs Ž .s ¨TTs Ž .Ž¨¨s T . Ž.3 IRtTtT@X# :I@X# :Ž.I@X# Tt. 1 1 Proof. Ž.1 For I a nonzero fractional ideal, IR@ X #y s Iy @ X # s I @ X #y1. A proof of this resultŽ due to D. F. Anderson and the second y1y1 author. may be found inwx 9, Proposition 2.1 . Thus ŽIR@ X # .TTs I @ X # y1 y1 y1 sŽI@X#.ŽTT. Because I @ X # .ŽI @ X #T: R@ X #TT, we have I @ X # . y1 :Ž.I@X#T . Now suppose that R is formally integrally closed and y1 T:NtT. Let 0 / g g Ž.I @ X # . Choose 0 / b g I.Sobg g R@ X #T. Write bg s frt00where f g R@ X # and t g T;sobgt0s f. Now for 0 / a g I, XX af s bagtŽ.0. Because ag g R@ X #T , ag s g rh where g g R@ X # and h g X T. Thus afh bg t and hence aA bA X. Because Ž.A R and R is s 0 fhs g t0 h t s formally integrally closed, aA aAŽ.aAA Ž .aA Ž .bA ŽX . fftfhtfhtgtt: s s s 0 y1 y1 :bR. Because 0 / a g I was arbitrary, A f : bI . Hence f g bI @ X # y1 y1 y1 y1 y1 so bfgI@X#. Thus g s bfrt0 gI@X#TT. Hence Ž.I @ X # : y1 ŽI@X#.T. Ž. Ž.Ž y1.Ž. Ž. 2 Part 1 replacing I by I yields IR@ X # ¨¨s I @ X # s I @ X # ¨ ŽŽ . . ŽŽ . . and hence IR@ X # ¨ T s I¨ @ X #T s I @ X # ¨ T . Because R is formally Ž . ŽŽ .y1 .y1 Ž y1 .y1 integrally closed and T : NtT, I @ X # ¨s I @ X #TTs I @ X # Žy1.y1 Ž.ŽŽ.. sI@X#T sI¨ @X#TT. Similarly, IR@ X # ¨¨s IR@ X # T. Ž.3 Using Ž. 2 , we get IRtt@X#:I@X#.Ž This is also given inw 9, Proposition 2.4x .. Thus IRtTtT@X# :I@X# . Now for any domain R, nonzero ideal J of R, and multiplicatively closed subset S of R we have Ž.AtS: Ž.ASt. Thus the last containment holds. PROPOSITION 3.5. Let R be a formally integrally closed domain. If J : Ž. R@X#is a t-ideal such that J l R / 0, then J s Ý0 / f g JfA ¨@ X #. 1 1 Proof. Let 0 / a g J l R and let f g J.If hgŽ.a,fy:aRy@X#, then hf g R@ X #. Because ah, f g R@ X # and R is formally integrally closed, Ž.ŽAahfts AA ahft .. Because aA Ž.Ž.Žhfts A ahfts AA ahft .= y1 y1 y1 aAhf A , we have Ah: A f, i.e., h g A f@ X #. Therefore Ž.a, f : Ž.y1 Ž. Žy1 .y1 ŽŽ .y1 .y1 A ff@ X #. By Theorem 3.4, A ¨@ X # s A f@ X # : a, f s Ž. Ž. a,f¨ :J.SoÝ0 / f g JfA ¨@X#:J. Because the reverse containment is obvious, we get equality. 354 ANDERSON AND KANG PROPOSITION 3.6. Let R be a formally integrally closed domain. If 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. Ž. Proof. Let J s f1,..., fn ¨ where f1,..., fn gR@X#. By Proposition 3.5, J Ý Ž.A @ X # ŽA .@ X # иии ŽA .@ X # ŽŽ.A s 0 / f g Jf¨ = f1¨ q q fn¨ s f1¨ иии Ž..A @ X #.Because J is a -ideal, J ŽŽŽ.A иии qq f n ¨ ¨ = f1 ¨ q Ž...A @X# ŽŽ.A иии Ž..A @ X # ŽA иии A .@ X #. qf n¨¨s f1¨qq fn¨¨ s ff1q q n¨ Obviously, J ŽŽ A иии A ..Ž@ X # A иии A . @ X # and there- : ff1q q n¨s ff1q q n¨ fore J Ž.A иии A @ X #. s ff1q q n¨ PROPOSITION 3.7. Let R be a formally integrally closed domain and let T : Nt be a multiplicati¨ely closed subset of R@ X #. Then the set S s Ä J : R@ X #TT< J is a proper t-ideal of R@ X # with J l R s 04 has maximal elements and each maximal element of S is a prime ideal. Proof. Because Ž.X T g S, S / л. Let Ä4J␣ be a chain in S. Clearly DJ␣ is a proper t-ideal of R@ X #T and Ž.D J␣␣l R s D ŽJ l R .s 0. So DJ␣ gS. By Zorn’s Lemma, S has a maximal element, say J. Suppose J is not a prime ideal of R@ X #T . Let Q␣ be a prime ideal of R@ X #T which is minimal over J. Then Q␣ is a prime t-ideal and Q␣ l R / 0 by the maximality of J in S. Let P␣ s Q␣␣l R@ X #Ž./ 0 . Note that P is a prime t-ideal of R@ X # with P␣ l R / 0.ŽŽŽ..ŽŽ..Ž. For P␣ tT: P␣Tts Q␣ts Q␣ and hence Ž.P␣ t : Q␣␣l R@ X # s P .. By Proposition 3.5, P␣s ÝI␣@ X # for a collection of ideals Ä4I␣␣of R. Thus Q s Ž.P␣Ts Ý␣I @ X #T. Now Q␣sŽ.Q␣ts ŽÝ␣I@X#Tt .s ŽŽÝ␣I@X#Ttt ..ŽŽ=Ý␣IR .tTt@X# .s ŽŽÝ ŽIR␣ .tTt .@X# .= ŽŽŽÝ␣ ŽIR .ttT .@X# . .= ŽŽÝ␣ ŽIR .tt .@X# . = ŽŽÝ IR␣ ..tt @X#= ŽŽÝ␣I ..ŽttsÝ␣I .t where the first containment relation follows from Theorem 3.4Ž. 3 . Thus ŽÝ I␣ .t : Q␣ and hence F␣Ž.Ý␣It:F␣␣Qs''J. Also, P␣: Ž.Ý ␣I t@ X #,so JsF␣␣Q s F␣Ž.ŽP␣TsF␣␣P .T: ŽŽŽ.F␣␣ÝItT@X# ..ŽŽsF␣␣ÝI ..tT@X# where the third equality follows from the fact that each P␣ is the contraction of Q␣ . Because 'J / 0, F ␣␣Ž.Ý I t/ 0. Hence 0 / n F␣Ž.Ý␣It:''JlR. Let 0 / a g J l R. Then 0 / a g J l R for some n ) 0 which contradicts J l R s 0. Therefore J is a prime ideal. 4. THE DOMAINS RXŽŽ ..AND RX ÄÄ 44 Let R be an integral domain with quotient field K. Recall that RXŽŽ .. R@ X # and RXÄÄ 44 R@X# where N Ä f R@ X #< A R4 and s NNs ts g fs NtfsÄfgR@X#<Ž.AtsR4. We first give the power series analog of Theorem 2.6. FORMALLY INTEGRALLY CLOSED DOMAINS 355 THEOREM 4.1. For an integral domain R with quotient field K, the following statements are equi¨alent. Ž.1 R is a Krull domain. Ž.2 RÄÄ X 44 is a Krull domain. Ž.3 RÄÄ X 44 is a PID. Ž.4 RÄÄ X 44 is a Euclidean domain. Proof. ClearlyŽ. 4 « Ž.3 « Ž.2 and Ž. 2 « Ž.1 because RXÄÄ 44 l Ž. Ž. ÄÄ 44 ÄÄ 44 Ž.ÄÄ 44 K s R.1 « 3 . We first show that fR X s AXffsAX¨s ŽÄÄ 44.ŽŽ.ÄÄ 44. U ARf X ¨ s ARXf ¨¨for each f g R@ X # . Because R is a Krull Žy1. y1 domain, AAff tsRand A fs I¨for some finitely generated frac- tional ideal I of R. Choose g g KXwxwith Agfs I. Now Ž.AAgtsR. U Choose b g R with bI : R. Because bg g RXwx,bAŽ.Žfg ts A fŽbg. .t s Ž.Ž.AAf bgtsbAA f gtsbR; and hence Ž.A fgts R. Thus fg g Nt. For a A,a fagŽ.fg where ag RX and fg N .Soa fR@ X # gfts r g wx g g Nts ÄÄ 44 ÄÄ 44 Ž ÄÄ 44. ÄÄ 44 fR X . Hence A ff: fR X and so AR XX ¨:fR X . By Theo- Ž.ÄÄ 44 Ž ÄÄ 44. ÄÄ 44 ÄÄ 44 Ž.ÄÄ 44 rem 3.4, AXf¨ sARf X ¨;so fR X : AXff:AX¨ ÄÄ 44 ÄÄ 44 ÄÄ 44 Ž.ÄÄ 44 Ž ÄÄ 44. : fR X . Hence fR X s AXffsAX¨sARf X ¨s ŽŽ . ÄÄ 44. ARXf¨¨. Ä4 Ž.Ä4 XÄ4 Now by Theorem 2.6, RX is a PID. So ARXf ¨ sfRX for some X Ž.ÄÄ 44 X ÄÄ 44 ÄÄ 44 f g RXwx.Hence ARXf ¨ sfR X and thus fR X s ŽŽ . ÄÄ 44.ŽXÄÄ 44. X ÄÄ 44 ARXf ¨¨sfR X ¨sfR X . So for each f g R@ X # there X X exists f g RXwxwith fRÄÄ X 44 s fRÄÄ X 44. So let fRÄÄ X 44and gR ÄÄ X 44 be two principal ideals of RXÄÄ 44. We may X assume that f, g g R@ X #. Now fRÄÄ X 44 s fRÄÄ X 44and gR ÄÄ X 44 s X XX X gRÄÄ X 44 for some f , g g RXwx. Because RXÄ4is a PID, fRXÄ4q X gRÄ4 X shR Ä4 X for some h g RXwx. Then fRÄÄ X 44q gR ÄÄ X 44 s X X X X fRÄÄ X 44 qgRÄÄ X 44 s Ž fRXÄ4qgRÄ4 X. RÄÄ X 44 sŽhRÄ4 X. RÄÄ X 44 s hRÄÄ X 44. Thus RX ÄÄ 44 is a Bezout domain. But because RXÄÄ 44 is a Krull domainŽ. it is a localization of a power series ring over a Krull domain , RXÄÄ 44 is a PID. Ž.3« Ž.4 . Bywx 10, Theorem 5.3 , it suffices to show that 1 is in the stable range of RXÄÄ 44, that is, if h12, h g RXÄÄ 44 with Ž.h12, hRXÄÄ 44 s RXÄÄ 44, then Ž.h12q uh RÄÄ X 44s RX ÄÄ 44 for some u g RXÄÄ 44. By choos- ing a common denominator for h12and h we can assume h12, h g R@ X #. Put ˆh hXŽ.2and ˆh Xh Ž. X 2and h ˆˆh h . Now A A and 11s 2s 2 s 12q hhˆiis A A A. Note that if f, g R@ X # with Ž.A Ž.A , then hhs12q h g f¨: g¨ ÄÄ 44 Ž.ÄÄ 44 Ž.ÄÄ 44 ÄÄ 44 ŽŽ.Ž.. fR X s AXf ¨ :AXg ¨ sgR X as shown in 1 « 3 and hence frg g RXÄÄ 44.SohiirhgRXÄÄ 44 and ˆh rhiis a unit in RXÄÄ 44. Thus hiis Ž.h rhhghRÄÄ X 44,sohR ÄÄ X 44 s Ž.h12, hRXÄÄ 44sRX ÄÄ 44. Then h s hh111Ž.Ž.ŽŽ.Ž..ˆˆrh qh 2rhh 22sh 1qh 1rˆˆˆhh 1 2rhhh 221rh 1 356 ANDERSON AND KANG where Ž.Ž.h11rˆˆhh 22rh and ˆh 11rh are units in RXÄÄ 44.SoŽh1q uh21. RÄÄ X 44s RX ÄÄ 44 where u s Ž.Ž.h rˆˆhh12rh2gRXÄÄ 44.Ž Note that we have actually given an alternative proof that RXÄÄ 44 is Bezout.. COROLLARY 4.2. Let R be a Krull domain. X Ž.1 For each f g R@ X # there exists f g Rwx X with fRÄÄ X 44 s fRX ÄÄ X 44. Ž. U ÄÄ 44 Ž.ÄÄ 44 Ž.ÄÄ 44 2 For f g R@ X # , fR X s AXf¨ sAXf s Ž ÄÄ 44. ARf X ¨. Ž. U ÄÄ 44 ÄÄ 44 Ž. Ž. 3For f, g g R@ X # , fR X s gR X m A f ¨ s Ag ¨ . Proof. These three statements are proved inŽ. 1 « Ž.3 of Theorem 4.1. Inwx 4, Theorem 2.13 we showed that if an integral domain R satisfies Ž. Ž . U Afg ¨ sAAfg¨for all f, g g R@ X # , then we can define the power ˆ ˆ series ring analog R¨of the Kronecker function ring R¨¨by R s Äfrgf<,g U Ž. Ž.4Ä4 ¨ˆ gR@X#with A f ¨ : Ag ¨ j 0 . Then R is a completely integrally ¨ˆ Ž.¨ˆ closed Bezout domain and for 0 / f g R@ X #, fR s ARf ¨s ŽŽ .¨ˆ . ÄÄ 44 ¨ˆ ARf¨¨. We next show that if R is a Krull domain, then RX sR. This gives an alternative proof that if R is a Krull domain, then RXÄÄ 44 is a PID. ˆ THEOREM 4.3. Suppose that R is a Krull domain. Then RÄÄ X 44 s R¨. Hence if R is a Krull domain, RÄÄ X 44 is a PID. Ž. Ž . Proof. Suppose that R is a Krull domain. Then A fg ¨ s AAfg¨for all f, g R@ X #U;so R¨ˆis defined. Clearly RXÄÄ 44 R@X# R¨ˆ. Con- g s Nt : ¨ˆ U Ž. Ž. versely, let 0 / frg g R where f, g g R@ X # with A f ¨ : Ag ¨ . Choose Ž. y1 Ž.Ž. hgKXwxwith Ah ¨ s Aggso R s AAhtsAght. Then frg s fhrgh where gh g Ntand ŽA fht .s ŽAA fht .s ŽŽAA ftht . .: ŽŽAA gtht . . s Ž.AAghtsR,so fh g R@ X #. Hence frg g RXÄÄ 44. According to Theorem 2.5 PVMD’s are characterized by the property that everyŽ. principal ideal of RXÄ4is extended from R. We next show that for RXÄÄ 44, this property characterizes Krull domains. To do this we need several lemmas. LEMMA 4.4. Let R be an integral domain. Let 0 / f g Rwx X and l g R@X#withŽ. Alts R. Then Ž A flt .s ŽA ft .. Proof. Now for ideals I and J of R,if IMMsJ for each M g t-MaxŽ.R , then IttMMJ . ŽI J gives IRwx X N IR wx X MXIRXMM wxX s s ¨MwXxNs wxs wxs ¨ JRXMMwxXJR wx X MXJR wx X NNNand hence IRwx X JR wx X wxs wxs ¨MwXxN ¨¨s ¨ because t-MaxŽRXwx. ÄMRwx X< M t-MaxŽ.R 4byw 17, Proposition NN¨¨s g FORMALLY INTEGRALLY CLOSED DOMAINS 357 2.1xw . By 17, Proposition 2.2 xwx , IRX ŽIRwx X .ŽJRwx X . tN¨¨s Nts Nt ¨s JRXwx and hence byw 17, Proposition 2.8x , I IRX wx R tN¨ ttNs ¨l s JRXwx R J..Ž.Ž. Hence it suffices to show that A A where tN¨l s t flMfMs Ž. иии Mgt-Max R . Let l s a01q aXq where a0,...,aiy1gM, but aif Ž . Ž .Ž.Ž. M. It is easily checked that A fl M q a0,...,aAiy1 MfMsA fM.By Nakayama’s Lemma, Ž.A fl Ms Ž.A f M. LEMMA 4.5. Let R be an integral domain. Let 0 / f g RXwx.If fRÄÄ X 44 is extended from R, then fRÄÄ X 44s ARf ÄÄ X 44. Proof. Let fRÄÄ X 44s IR ÄÄ X 44. We can take I s Ž.i1,...,in to be finitely generated. Then f s if11rlqиии qifnnrlfor some f ig R@ X # and l g N t. Then fl s if11qиии qifnn,soŽ.A fts ŽA flt .:I t by Lemma 4.4. So fRÄÄ X 44 s Ž fRÄÄ X 44.Žttts IRÄÄ X 44.Žs IXÄÄ 44. =IRtfÄÄ X 44=AR ÄÄ X 44 = fRÄÄ X 44 where we have used Theorem 3.4 and that I is finitely generated. LEMMA 4.6. Suppose that e¨ery ideal of RÄÄ X 44 is extended from R. Then R is formally integrally closed. Proof. It suffices to show that RXÄÄ 44 is Bezout. For if M g t-MaxŽ.R , then R@ X # M @ X # is a localization of RXÄÄ 44 and hence is a valuation domain. But then bywx 8, Theorem 1 , RM is a DVR. Hence by Proposition 3.3, R is formally integrally closed. To show that RXÄÄ 44 is Bezout, it suffices to show that for f12, f g U R@ X # , fR12ÄÄ X 44qfR ÄÄ X 44 is principal. Let fRiiÄÄ X 44sIR ÄÄ X 44 where X Iis a finitely generated ideal of R. Choose f RX with A X I . Then iig wx fis i fR1212ÄÄ X 44qfR ÄÄ X 44sIR ÄÄ X 44qIR ÄÄ X 44 sŽ.I 12qIRXÄÄ 44 s X n X X ARXXXnÄÄ 44 Ž fXf. RÄÄ X 44 by Lemma 4.5 where n ) deg f . Ž f12qXf. s12q 1 THEOREM 4.7. For an integral domain R, the following conditions are equi¨alent. Ž.1 R is a Krull domain. Ž.2 E¨ery principal ideal of RÄÄ X 44 is extended from R. Ž.3 E¨ery ideal of RÄÄ X 44 is extended from R. Ž.4 R is formally integrally closed and e¨ery prime ideal of RÄÄ X 44 is extended from R. Ž.5 R is formally integrally closed and e¨ery prime t-ideal of RÄÄ X 44 is extended from R. Proof. Ž.1 « Ž.2 . Let 0 / f g R@ X #. By Corollary 4.2, there exists X X f g RXwxwith fRÄÄ X 44 s fRÄÄ X 44. Now because R is a Krull domain and X hence a PVMD, by Theorem 2.5 fRXÄ4sAR Ä4 X for some ideal A of R. 358 ANDERSON AND KANG X X Thus fRÄÄ X 44 s fRÄÄ X 44 sŽ fRXÄ4. RÄÄ X 44sAR ÄÄ X 44. So every princi- pal ideal of RXÄÄ 44 is extended from R.2Ž.m Ž.3 . This holds for any ring R.3Ž.« Ž.4 . We only need show that R is formally integrally closed and this follows from Lemma 4.6.Ž. 4 « Ž.5 . Clear. Ž. 5 « Ž.1 . By Proposition 3.7, J l R / 0 for each t-ideal J of RXÄÄ 44. In particular, fRÄÄ X 44 l R / 0 U for each f g R@ X # . Let 0 / a g fRÄÄ X 44 l R. Hence a s fhrl for some U h,lgR@X# with Ž.Alts R.Soal s fh and hence aR s aAŽ.lts Ž.Ž.ŽAal tsA fh tsAA f h . t because R is formally integrally closed. Thus Af is t-invertible. So every countably generated nonzero ideal of R is t- invertible. We show that every nonzero ideal of R is t-invertible. Let J be a t-ideal of R.If J/It for any finitely generated ideal I : J, we get an infinite ascending chain Ž.a1 t ; Ža12, a .t; Ža123, a , a .t; иии where each Ä4 ϱ Ž.ŽϱŽ.. anngJy0 . Now D s11a ,...,ants D ns11a ,...,antis a t-ideal of ϱ Ž. countable type and hence is t-invertible. Thus D ns11a ,...,ants Ž.a1,...,amt for some m Žbecause a t-invertible ideal has finite type. , a contradiction. Hence every t-ideal J of R has the form J s It for some finitely generated ideal and hence is t-invertible. Thus R is a Krull domain. Now in general, we need not have IXÄÄ 44sIR ÄÄ X 44 Žwhere of course IXÄÄ 44 I@X#.. We say that an ideal J of RXÄÄ 44 is formally extended s Nt from R if J s IXÄÄ 44 for some ideal I of R. We have the following companion result to Theorem 4.7. THEOREM 4.8. For an integral domain R, the following conditions are equi¨alent. Ž.1 R is a Krull domain. Ž.2 R is formally integrally closed and e¨ery principal ideal of RÄÄ X 44 is formally extended from R. Ž.3 E¨ery ideal of RÄÄ X 44 is formally extended from R. Ž.4 R is formally integrally closed and e¨ery prime ideal of RÄÄ X 44 is formally extended from R. Ž.5 R is formally integrally closed and e¨ery prime t-ideal of RÄÄ X 44 is formally extended from R. Proof. Ž.1 « Ž.2 . Corollary 4.2. Ž. 2 « Ž.1 . Let 0 / f g R@ X #. Now fRÄÄ X 44s IX ÄÄ 44 for some ideal I of R,so fRÄÄ X 44 l R / 0. Now as in the proof ofŽ. 5 « Ž.1 of Theorem 4.7, we see that R is a Krull domain. Ž.1 « Ž.3 . By Theorem 4.1, RXÄÄ 44 is a PID. Thus each ideal of RXÄÄ 44 has the form fRÄÄ X 44 for some f g RXÄÄ 44 and hence is formally extended by Corollary 4.2.Ž. 3 « Ž.1 . We first show that R is a PVMD. Let 0 / f g RXwxand let I be an ideal of R such that fRÄÄ X 44s IX ÄÄ 44. FORMALLY INTEGRALLY CLOSED DOMAINS 359 Using Lemma 4.4, it is easily shown that Ž.A fts I t. Now fRÄÄ X 44 s ŽfRÄÄ X 44.Žttts IXÄÄ 44. =IXÄÄ 44 by Theorem 3.4. So fRÄÄ X 44s IXt ÄÄ 44 s Ž.AXftÄÄ 44=AXf ÄÄ 44=fR ÄÄ X 44 and hence fRÄÄ X 44s AXf ÄÄ 44. Thus RXÄÄ 44fR ÄÄ X 44Ž fRÄÄ X 44.y1 AXAXÄÄ 44Ž ÄÄ 44.y1 A@X#Ž A@X#y1. s s ffs fNfNtt A @ X # Ay1@ X # ŽAAy1.@X# RXÄÄ 44 where the third equality s fNfNtt: ffNt: follows because A @ X # is finitely generated. Thus ŽAAy1.@X# fffNts R@X#,soŽAAy1.R. Thus R is a PVMD. Next we show that every Nfftt s principal ideal of RXÄÄ 44 is extended from R. Let 0 / f g R@ X #, let I be an ideal of R such that fRÄÄ X 44s IX ÄÄ 44, and let J be an ideal of R with IRÄÄ X 44s JX ÄÄ 44. It is easy to check that Itts J . Hence fRÄÄ X 44 s ŽfRÄÄ X 44.Žttts IXÄÄ 44. =IXÄÄ 44=fR ÄÄ X 44. Thus fRÄÄ X 44s IXt ÄÄ 44 s JXttttÄÄ 44 :ŽJXÄÄ 44.ŽsIRÄÄ X 44.Ž: IXÄÄ 44. sfRÄÄ X 44 and hence fRÄÄ X 44 X s ŽIRÄÄ X 44.t. Let I : I be a finitely generated ideal of R with f g X X Ž IRÄÄ X 44.tt.So fRÄÄ X 44 s ŽIRÄÄ X 44. . Now because R is a PVMD, by Theorem 2.5 RXÄ4is a Bezout domain and hence IRXXÄ4is a principal X X ideal. So IRÄÄ X 44 sŽIRXÄ4. RÄÄ X 44 is principal. Thus fRÄÄ X 44 s X X Ž IRÄÄ X 44.t sIRÄÄ X 44. Hence each principal ideal of RXÄÄ 44 is extended from R. By Theorem 4.7 R is a Krull domain.Ž. 1 « Ž.4 . This is the same asŽ. 1 « Ž.Ž.3. 4 « Ž.5 . Clear. Ž. 5 « Ž.1 . This is the same as the proof ofŽ. 5 « Ž.1 of Theorem 4.7. COROLLARY 4.9. Let R be a Krull domain. Then MaxŽ RXÄÄ 44. s Ž1. ÄÄÄ44MX< MgXRŽ.4. Ž1. Proof. Because Nts R@ X # y DÄM @ X # THEOREM 4.10. For an integral domain R the following conditions are equi¨alent. Ž.1 R is a Dedekind domain. Ž2 .E¨ery principal ideal of R ŽŽ X .. is extended from R. Ž3 .R is formally integrally closed and e¨ery principal ideal of R ŽŽ X .. is formally extended from R. Ž4 .E¨ery ideal of R ŽŽ X .. is Ž formally . extended from R. Ž5 .R is formally integrally closed and e¨ery prime ideal of R ŽŽ X .. is Ž.formally extended from R. 360 ANDERSON AND KANG Ž6 .R is formally integrally closed and e¨ery prime t-ideal of R ŽŽ X .. is Ž.formally extended from R. U Ž.7 Aisinf ¨ertible for e¨ery f g R@ X # . Ž8 .R ŽŽ X .. is a Dedekind domain. Ž9 .R ŽŽ X .. is a PID. Ž10 .R ŽŽ X .. is a Euclidean domain. Proof. Observe that if dim R s 1, A ffs R m Ž.A ts R and hence NsNt. Thus RXŽŽ .. sRXÄÄ 44. HenceŽ. 1 « Ž.2 follows from Theo- rem 4.7 andŽ. 1 « Ž.3 , Ž. 4 follows from Theorem 4.8. For the extended caseŽ. 2 « Ž.4 « Ž.5 « Ž.6 follows as in Theorem 4.7. Also, for Ž. 4 in the formally extended case we get that for each ideal I of R, IRŽŽ X .. s JXŽŽ ..for some ideal J. Then I s IR ŽŽ X ..l R s JX ŽŽ .. lRsJ,so IRŽŽ X ..s IX ŽŽ ..for each ideal I of R. Hence if each ideal of RX ŽŽ .. is formally extended from R it is also extended from R. Hence by the proof of Lemma 4.6, R is formally integrally closed. SoŽ. 4 « Ž.5 « Ž.6 for the U formally extended case also.Ž.Ž. 3 , 6 « Ž.7 . Let f g R@ X # . By Proposi- tion 3.7, fRŽŽ X ..l R / 0. Hence by Proposition 3.5 fR ŽŽ X .. l R@ X # s Ý␣ I␣␣@X#for some collection Ä4I of ideals of R.ŽŽ Note that fRŽ. X .l R@X#is a t-ideal of R@ X #.. Because fR ŽŽ.. X is principal, fRŽŽ.. X s IXŽŽ ..иии IX ŽŽ .. ŽÝn IX.ŽŽ .. for some finite subcollection ␣ qq ␣ nisis1 ␣ Ä4I .So fRŽŽ X ..IX ŽŽ .. for some ideal I of R. It is easily checked that ␣ i s Ž . Ž .ŽŽ .. Ž . ŽŽ .. ŽŽ .. ŽŽ .. Ž ŽŽ ... Itftfs A . Now AX:AX fttsIX:IX¨¨sIX Ž ŽŽ ... ŽŽ .. Ž .ŽŽ .. ŽŽ .. ŽŽ .. s fR X ¨s fR X : AXff. Thus fR X s AX.So y1y1 y1 ŽAAff.ŽŽ X ..=AXAf ŽŽ .. fŽŽ X ..sAXAXf ŽŽ .. f ŽŽ .. sfRŽŽ X .. 1 ŽfR ŽŽ X ...y s RXŽŽ .. Žwhere the first equality follows from Theorem 3.4. . y1 Hence AAffsRand A fis invertible.Ž. 7 « Ž.1 . Clear. Ž. 1 « Ž10 . . Here RXŽŽ .. sRXÄÄ 44 is a Euclidean domain by Theorem 4.1.Ž. 10 « Ž.9 « Ž.8 . Clear. Ž. 8 « Ž.1 . Because R s RX ŽŽ..lK,Ris a Krull domain. Let I be a nonzero ideal of R. Because R is formally integrally 1 1 1 closed, IXŽŽ ..y sIXy ŽŽ ...Thus RXŽŽ ..sIX ŽŽ ..Ž IX ŽŽ ...y s 1 1 1 IXIŽŽ .. y ŽŽ X ..: Ž IIy .ŽŽ X ..: RX ŽŽ ... Thus RX ŽŽ ..s Ž IIy .ŽŽ X .. and 1 hence IIy s R;so Iis invertible. Hence R is a Dedekind domain. EXAMPLE 4.11. In Theorems 4.7Ž.Ž. 4 , 5 , 4.8 Ž.Ž. 4 , 5 , and 4.10 Ž.Ž. 5 , 6 , the condition that R be formally integrally closed cannot be omitted. Let R be a one-dimensional Noetherian domain that is not integrally closed. Then RXÄÄ 44 sRXŽŽ .. is a one-dimensional Noetherian domain whose maximal ideals have the form MRÄÄ X 44s MX ÄÄ 44 for M g MaxŽ.R . Thus every prime ideal of RXÄÄ 44 sRXŽŽ .. is extended and formally extended from R, but R is not formally integrally closed. FORMALLY INTEGRALLY CLOSED DOMAINS 361 Also, if V is an n-dimensional discrete valuation domain, then every prime ideal of VXÄÄ 44 sVXŽŽ .. is formally extended from V Žw7, Corollary 3.6x.Ž , but V is formally integrally closed m n s 1wx 4, Theorem 2.4. . According to Theorem 2.5 if R is a PVMD, then LtŽ.R ( L ŽRXÄ4. under the canonical lattice homomorphism. We next show that if R is a Krull domain, then LtŽ.R ( L ŽRXÄÄ 44.. In particular, L ŽR .( L ŽRX ŽŽ ... if R is a Dedekind domain: THEOREM 4.12.Ž. 1 If R is a Krull domain, then Lt ŽR .( L ŽRXÄ4.( LŽRXÄÄ 44.. Ž.2 If R is a Dedekind domain, then L ŽR .( L ŽRX Ž ..(L ŽRX ŽŽ .... Proof. Ž.2 follows from Ž. 1 because if R is a Dedekind domain, then RXŽ.sRXÄ4and RXŽŽ .. sRXÄÄ 44. So suppose that R is a Krull domain. The isomorphism LtŽ.R ( L ŽRXÄ4.is given by Theorem 2.5. Define : LtŽ.R ª L ŽRXÄÄ 44.Ž.by A s ARÄÄ X 44. Note that because the ¨ and t operations coincide on R and because every ideal of RXÄÄ 44 is a t-ideal, by Theorem 3.4 we get ARÄÄ X 44s AX ÄÄ 44sAXtt ÄÄ 44sAR ÄÄ X 44 for any nonzero ideal A of R. Hence Ž.ŽŽ..Ž.A( tttB s AB s AB RÄÄ X 44 s ABRÄÄ X 44s AR ÄÄ X 44 BR ÄÄ X 44 s Ž.Ž.A B and ŽA k t B .s ŽŽA q B .tt .s ŽA q BRX . ÄÄ 44 sŽ.AqBRÄÄ X 44sAR ÄÄ X 44q BR ÄÄ X 44 Ž1. Žn1. sŽ.Ak Ž.B. Note that for P1,...,Pn gXRŽ.we have ŽP1 Ž n s. n1 ns n1 ns n1 l иии l Ps .ŽŽs P1 иии Pst..s ŽP1 иии PRXst. ÄÄ 44 sP1 иии ns n1 ns PRXs ÄÄ 44 sŽPR1 ÄÄ X 44. иии ŽPRs ÄÄ X 44. . Because ÄÄÄ44ÄÄ44PR X s PX< P Ž1. gXRŽ.4is the set of maximal ideals of RXÄÄ44and RX ÄÄ44is a PID this shows that is a bijection. The fact that preserves meets easily follows Ž n1.Žns.Žm1.Žms. from the fact that Ž P1 l иии l Ps .Žl P1 l иии l Ps .s ŽmaxÄn11, m 4. ŽmaxÄnss, m 4. P1 lиии l Ps . 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