Class Semigroups of Prufer Domains

JOURNAL OF ALGEBRA 184, 613᎐631Ž. 1996 ARTICLE NO. 0279

Class Semigroups of Prufer¨ Domains

S. Bazzoni*

Dipartimento di Matematica Pura e Applicata, Uniërsita´ di Padoä, ïa Belzoni 7, 35131 Padua, Italy

Communicated by Leonard Lipshitz

View metadata, citation and similar papers at Receivedcore.ac.uk July 3, 1995 brought to you by CORE provided by Elsevier - Publisher Connector

The class semigroup of a commutative integral domain R is the semigroup S Ž.R of the isomorphism classes of the nonzero ideals of R with the operation induced by multiplication. The aim of this paper is to characterize the Prufer¨ domains R such that the semigroup S Ž.R is a Clifford semigroup, namely a disjoint union of groups each one associated to an idempotent of the semigroup. We find a connection between this problem and the following local invertibility property: an ideal I of R is invertible if and only if every localization of I at a maximal ideal of Ris invertible. We consider the Ž.࠻ property, introduced in 1967 for Prufer¨ domains R, stating that if ⌬12and ⌬ are two distinct sets of maximal ideals of R, Ä 4Ä 4 then F RMM

INTRODUCTION

Rwill denote a commutative domain. Let FŽ.R be the set of nonzero fractional ideals of R; FŽ.R is a commutative semigroup under multiplication and contains the group PŽ.R consisting of the principal ideals. The factor semigroup S Ž.R s F Ž.R rP Ž.R is the commutative semigroup of the isomorphism classes of nonzero integral ideals of R with the operation induced by multiplication. We define S Ž.R to be the class semigroup of R. SŽ.Rcontains as a subgroup the class group of R defined as the factor group IŽ.R rP Ž.R , where I Ž.R denotes the set of the invertible fractional

*Research supported by MURST. E-mail: [email protected].

613

0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 614 S. BAZZONI ideals of R. While the class group has been studied for various classes of integral domains, the semigroup S Ž.R has not received as much attention: it has been investigated by Salce and the author inwx 2 and by Zanardo and Zannier inwx 12 but only for particular classes of domains. In wx 2 the case of a valuation domain R is considered and the structure of the semigroup SŽ.Ris completely determined: it is proved that S Ž.R is a Clifford semigroup, namely a disjoint union of groups associated to the idempo- tents of the semigroup. Inwx 12 the class semigroup of some orders in algebraic number rings is investigated; moreover it is pointed out that if R is an integrally closed domain with Clifford class semigroup then R has to be a Prufer¨ domain. The aim of this paper is to solve the problem, raised inwx 2 , namely to give a characterization of the class of the Prufer¨ domains R for which SŽ.Ris a Clifford semigroup. We recall that a commutative semigroup S is a Clifford semigroup if every element a of S is regularŽ. von Neumann regular , namely if 2 a s a x for some element x g S. In Section 1 we observe that S Ž.R is a Clifford semigroup if and only if 2 2 I s IIŽ :I. for every ideal I of R. In Section 2 we characterize the prime ideals P of the Prufer¨ domain R such that P, the isomorphism class of P, is a regular element of S Ž.R : they are either idempotent or maximal and invertible as ideals over their endomorphism ring P : P. The regularity of the elements P for every prime ideal P of R is not enough to guarantee that S Ž.R is a Clifford semigroup as is shown by an exampleŽ. see Example B after Proposition 2.7 . We give some necessary conditions satisfied by an ideal I of R such that its isomorphism class I is a regular element of S Ž.R and we prove that a sufficient condition on a Prufer¨ R, in order for S Ž.R to be a Clifford semigroup, is that every nonzero element is contained only in a finite number of maximal ideals. In Section 3 we investigate the structure of the idempotent associated to a regular element of S Ž.R ; this helps in characterizing the ideals I of R such that I is regular and allows us to prove that if S Ž.R is a Clifford semigroup, then R is such that the invertibility is a local property, namely Rsatisfies the following property which we denote by Ž.) : Ž. )An ideal I of R is invertible if and only if every localization IRM of I at a maximal ideal M of R is invertible. In Section 4 we study the class of Prufer¨ domains satisfying propertyŽ. * :. Clearly a Prufer¨ domain has propertyŽ. * if and only if every ideal that is locally invertible at each maximal ideal is finitely generated. A Prufer¨ domain such that every nonzero element is contained only in a finite number of maximal ideals satisfies propertyŽ. * . We prove that this CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 615 condition is also necessary if R is such that RM is finite-dimensional for each maximal ideal M or if R satisfies the Ž.࠻ property defined inwx 9, 10 as follows: the Prufer¨ domain R satisfies the Ž.࠻ property if and only if for every two distinct sets ⌬12and ⌬ consisting of maximal ideals of R, Ä 4Ä 4 Ž. FRMM

If P is a prime ideal and P n Q, there exists a finitely generated ideal I such that P n I : Q. We end Section 4 with the following conjecture: Conjecture. A Prufer¨ domain satisfies propertyŽ. * if and only if every nonzero element is contained only in a finite number of maximal ideals. By the results proved in Section 4 we are able to characterize the Prufer¨ domains R with Clifford class semigroup in case R satisfies one of the following conditions:

Ž.1 RM is finite-dimensional for each maximal ideal M of R. Ž.2 Rhas property Ž.࠻ . In any of the preceding cases R has Clifford class semigroup if and only if every nonzero element of R is contained only in a finite number of maximal ideals of R. Notice that this characterization would be valid in the general case if the above conjecture were proved true.

1. PRELIMINARIES

1.1 Commutati¨e Semigroups. Let S be a commutative semigroup with 1. Consider the set IŽ.S of the idempotent elements of S,

2 I Ž.S s Ä e g S : e s e 4 . 2 An element a in S is said to be regular Ž.¨on Newmann regular if a s ax for some element x g S. 616 S. BAZZONI

Let RŽ.S be the subsemigroup of the regular elements of S. Then RŽ.Sis a disjoint union of groups. In fact, for every idempotent e g I Ž.S , let Ge be the largest subgroup of RŽ.S containing e, namely

Ges Ä4ae< abe s e for some b g RŽ.S . Ž. Ä4 Then R S is the disjoint union of the family GeegIŽS.. Hence, if a g RŽ.S there exists a unique idempotent e such that Ž a g Ge. We say that e is the idempotent associated to aesax, where 2 a s ax.. Note that G1 is the subgroup of the invertible elements of S.If e,fare in IŽ.S and ef s e, then the multiplication by e induces a group homo- f morphism: ␾ef: G ª G ecalled the bonding homomorphism between Gf and Ge. DEFINITION 1. A commutative semigroup S is a Clifford semigroup if and only if S s RŽ.S . 1.2. Regular Elements of SŽ. R for a Commutati¨e Domain R. Let R be a commutative integral domain. Denote by S Ž.R the class semigroup of R consisting of the isomorphism classes of nonzero fractional ideals of R. If I is an ideal of R, denote by I its isomorphism class. Then

SŽ.RsÄ4IN0-IFR and I s J, for ideals I and J of R if and only if I s qJ for some element qgQ.Iis a regular element of S Ž.R if and only if there exists a 2 fractional ideal X of R such that I s I X. We start with some properties of the regular elements of S Ž.R . 22 LEMMA 1.1. I is a regular element of S Ž.R if and only if IŽ. I: I s I. 2 Proof. Assume I is a regular element of S Ž.R . Then IXsIfor 2222 some fractional ideal X of R. Hence X : I : I and I s IX:IIŽ:I. 2 2 2 :Iimplies I s IIŽ:I.. The converse is clear since I : I is a fractional ideal of R.

2 LEMMA 1.2. Let I be an ideal of R. If I is in¨ertible or isomorphic to I , then I is a regular element of S Ž.R . 2 Proof. If I is invertible, then IRŽ.:I sR; hence IRŽ.:IsI.If 2212 2 12 IsqI for some q g Q, then I : I s qIyŽ.:I; hence IIŽ:I.sqIy sI. The overring I : I will be denoted by D; I is clearly a D ideal.

LEMMA 1.3. In the abo¨e notations, if I is in¨ertible as a D ideal then I is a regular element of S Ž.R . CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 617

2 Proof. We have I : I s D : I.If Iis invertible over D, then IDŽ.:I s 22 D. Hence IIŽ.:IsID s I and we conclude by Lemma 1.1. 1.3 Notations. R denotes a commutative Prufer¨ domain and Q its quotient field. An overring of R is a ring between R and Q. If A and B are R-submodules of Q, A : B is defined as follows: Q A:BÄ4qQqB< A . Q sg: We will often omit the subscript Q in case no ambiguity arises. For every nonzero ideal I of R we consider the following objects:

y1 the fractional ideal R: I s IR ; the overring I : I, canonically isomorphic to EndŽ.I ; and the Nagata transform:

TIŽ. ŽR:In .. R sD ngގ MaxŽ.R and Spec Ž.R denote, respectively, the sets of the maximal and of the prime ideals of R, while MRŽ.,I is the subset of Max Ž.R consisting of the maximal ideals of R containing I. Inwx 5 , an explicit representation of I : I as the intersection of localizations of R is given, namely it is proved that

Ž.a I : I R R , s ž/ž/FFGŽM.l M MgMRŽ.,IMfMRŽ.,I where, for every M g MRŽ.Ž.,I,GM denotes the unique prime ideal of R Ž. such that GMRMMis the set of zero divisors of R rIRM. ࠻ Remark 1. GMRŽ.MMcoincides with the prime ideal Ž.IR associated to IRM , defined inwx 6 as follows: for every ideal J of a valuation domain V

࠻ JsÄ4rgVrJ< nJ . Inwx 1 it is also noticed that, for every ideal J

࠻ 1 J s D ry J rgV_J and

J s JR J ࠻ . Hence

࠻ GMRŽ.MMs ŽIR . 618 S. BAZZONI and IRMGs IR ŽM.. Ä Ž. 4 ŽŽ.. The overring F RMM

2. REGULAR ELEMENTS OF SRŽ. FOR A PRUFER¨ DOMAIN R

We are interested in characterizing the domains R for which the semigroup S Ž.R is a Clifford semigroup. As mentioned in the Introduc- tion, inwx 12 it is proved that if R is an integrally closed domain, then S Ž.R is a Clifford semigroup if and only if R is a Prufer¨ domain. Hence from now on we will assume that R is a Prufer¨ domain. We will frequently use the equivalent condition that RM is a valuation domain for every maximal ideal M g MaxŽ.R and also the fact that the prime ideals of R contained in a fixed maximal ideal M are linearly ordered by inclusion. D s I : I will denote the Prufer¨ overring of R given 2 by Eq.Ž. a . Let L s II Ž :I.ŽsID:I .. Then L is a D ideal and clearly IL : I. PROPOSITION 2.1. Let I be an ideal of R and assume I is a regular element of S Ž.R . Then the idempotent associated to I is L, where L s IDŽ.:I satisfies the following properties: Ž.1 L is an idempotent ideal of D and IL s I. Ž.2 DsI:IsL:LsD:L. Proof. Ž.1 By Lemma 1.1, the idempotent of S Ž.R associated to I is IDŽ.:I Ž.Ž.see Section 1.1 . Clearly L s ID:I is an idempotent ideal of Dand IL s I, provingŽ. 1 . Ž.2 Since L is idempotent, it is also a radical ideal of D, as noticed inwx 7, Sect. 23 . Now D s I : I s I : IL and I : IL s Ž.I : I : L s D : L. Hence D : L is a ring and thus, bywx 11, Prop. 3.9 , D : L s L : L. PROPOSITION 2.2. Let I be an ideal of R and assume I is regular. The following hold. 2 Ž.1 If D : I is a ring, then D : I s D s R : I and I s I . Ž. 2 Ž. Ž. 2 If I n I, then D n D : I n TIDRsTI. y1 Proof. Ž.1 Denote D : I by ID ; if that is a ring, then by Proposition 2.2 y1 y1 ofwx 11 , D : I s IIDD: II s L : L and by the preceding proposition L : L sD.D:IsDsI:Iclearly implies R : I s D. Moreover, since I is 222 regular, I s Ž.D : II sDI s I , hence we conclude. CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 619

2 Ž.2IfInI, then byŽ. 1 , D : I cannot be a ring, hence D n D : I.By Ž. Ž. the same observation D n TIDD, since D is clearly contained in TI. Ž. Ž. nnq1 nq1 Now, obviously TIRD:TI and D : I s I : I : R : I yields Ž. Ž. TIDR:TI. We list here some known facts about prime ideals in a Prufer¨ domain. Fact 1wx 11, Corollary 3.4 . If M is a maximal ideal of R, then M is invertible or R : M s M : M s R. Fact 2wx 11; 3, Lemma 3.0 . If P is a nonmaximal prime ideal of R, Ä Ž. 4 then R : P s P : P s RPMl RM< gMax R , M W P and P is a prime ideal of P : P. Ž.Ž Fact 3wx 3, Theorem 3.1 and Remark 3.2 . P : P n TPR the Nagata transform of P. if and only if P is a maximal invertible ideal of P : P. We now give a characterization of the prime ideals P of R such that P is a regular element of S Ž.R . Another characterization will be given in the 2 sequel. Notice that, by Lemma 1.2, it is enough to assume P n P . PROPOSITION 2.3. Let P be a nonidempotent prime ideal of R. These are equi¨alent: Ž.1 P is regular in S Ž.R . Ž.2 P is a maximal in¨ertible ideal of P : P. Ž. Ž . 3 P:PnTPR . Proof. Ž.1 « Ž.2 . Assume first that P is a maximal ideal of R; we prove that P is invertible. Suppose the contrary, then by Fact 1 we have 2 2 2 2 P:PsŽ.P:P:PsR:PsP:PsR. Hence PPŽ:P.sPnP contradicting the regularity of P. Now let P be a nonmaximal prime ideal of R and consider D s P : P. We first prove that P is a maximal ideal of 2 2 D. Assume the contrary, then by Fact 2 D : P s D, hence PPŽ :P. s 2 2 PDŽ.:PsPnP, a contradiction. It remains to show that P is invertible over D; if not, then by Fact 1 D : P s P : P s D and we can argue as before to get a contradiction. Ž.2« Ž.1 . Follows by Lemma 1.2. Ž.2« Ž.3 . This is Fact 3. COROLLARY 2.4. If M is a maximal ideal of R, then M is regular if and only if M is either idempotent or finitely generated. Proof. Clear by Proposition 2.3 and Fact. 1.

EXAMPLE A. Let R be the ring of entire functions. Then every free maximal ideal of R is neither idempotent nor finitely generated. Hence by Corollary 2.4, the class semigroup of R is not Clifford. 620 S. BAZZONI

We want now to characterize the regular elements I of S Ž.R for an arbitrary ideal I of R. Without loss of generality, we may assume I is not an idempotent ideal. With the same arguments used inwx 5 to prove Eq.Ž. a , we obtain the representation of I : I 2,

Ž.b I : I 2 IR : I 2 R C Ž.I , s FŽ.GŽM.GŽM.l C MgMRŽ.,I where CŽ.I R and GMŽ.denotes the unique prime ideal C s F M f MŽ R, I . M Ž. Ž .࠻ of R such that GMRMMs IR . Ž.࠻ We remark that, if P is a prime ideal, then PRMMPs PR s PR for every M g MRŽ.,P and thus:

2 2 Ž.c P : P s Ž.PRPP: P R l C Ž.P .

The following two lemmas are easy generalizations of similar results proved inwx 7 for ideals of a Prufer¨ domain. The first one holds over an arbitrary domain.

EMMA L 2.5. Let A be an R submodule of Q. Then A s F M g MaxŽR. ARM. Proof. The proof goes exactly as the proof of Theorem 4.10Ž. 3 inwx 7 .

LEMMA 2.6. Assume A and B are R submodules of Q and I is an ideal of R.ThenŽ. A l BIsAI l BI. Proof. In view of the preceding lemma, the proof is the same as the proof of Theorem 25.2Ž. c inwx 7 . We may now improve Proposition 2.3 by adding new equivalent conditions.

PROPOSITION 2.7. Let P be a nonidempotent prime ideal of R. These are equi¨alent: Ž.1 P is regular in S Ž.R . Ž.2 R R , where M is a maximal ideal of R. PMWF WPM Ž.3 There exists a finitely generated ideal I contained in P such that any maximal ideal of R containing I contains P. 2 Ž 2 . 2 Ž 2 . Proof. We first notice that PPRPP:PR sPR PPP PR :PR . But RP is a valuation domain and thus, as proved inwx 2 , S Ž.RP is a Clifford 2 Ž 2 . semigroup; hence PRPP PR :PR PsPR Pby Lemma 1.1. 2 2 Ž.1« Ž.2 . By hypothesis PPŽ :P.ŽsP; hence by Eq. c. and Lemma 2 Ž 2 . 2 Ž. 2.6, PPRPP:PR lPCPsP. CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 621

Ž. By way of contradiction assume that RP = C P . By Fact 2, we would Ž. Ž. Ž . Ž. get P : P s RP l C P s C P ; but PP:PsP, and hence P C P s 2 2 P. Now, as proved above, we obtain P s PRPl P ; hence P s P ,a contradiction. Ž. Ž. Ž. Ž. Ž. 2«1 . By Theorem 26.1wx 7 , C P RP implies P C P s C P ; 2 Ž 2 . 2 Ž. Ž. thus PP:PsPRPPl P C P s PR l C P which is P by Lemma 2.5. Ž.2m Ž.3 . Is Proposition 1.4 inwx 10 . One could ask whether the regularity of P for every prime ideal P of R implies that the class semigroup of R is a Clifford semigroup. The answer is negative as the following example shows.

EXAMPLE B. Let R s ޚ q XޑwwX xx where ޚ is the ring of integers and ޑ the field of rationals. The maximal ideals of R are principal and the only nonmaximal prime ideal is P s XޑwwX xx which is isomorphic to its square; hence P is regular. But the class semigroup of R is not Clifford. In Ä4 fact let I be the ideal generated by the set Xrp p where p is a prime of ޚ; 2222 then I : I s ޑwwX xx and thus IIŽ.:IsXޑwwX xx nIshowing that I is not a regular element of S Ž.R . The next statement simplifies the characterization of regular elements in SŽ.R.

LEMMA 2.8. Let I be a nonidempotent ideal of R. The following are equi¨alent: Ž.1 I is a regular element of S Ž.R . Ž. 2Ž 2.Ž.2 2IRMM: II:I R for e¨ery ideal M g MR,I such that I RM nIRM .

Proof. By Lemmas 1.1 and 2.5, I is regular if and only if IRM s 2 Ž 2 . 2 Ž 2 .Ž. II:IRMMMor equivalently IR : II:IR for every M g Max R . If M W I then IRMMs R and thus the above inclusion is verified since 2 I:I=R. 2 If M = I, but I RMMs IR , then again the inclusion follows. Thus Ž.1m Ž.2 is clear. We need a technical lemma on valuation domains.

LEMMA 2.9. Let V be a ¨aluation domain, J an archimedean ideal of V 2 2 such that J n J, and let P be a prime ideal of V not maximal. Then J VP = J if and only if J q P. ࠻ Proof. Recall that J is archimedean if J s M where M is the maximal ideal of V; hence J cannot be equal to P.If JqPthen 2 JVPPs V and hence J VPPs V = J. 622 S. BAZZONI

2 ࠻ Conversely, assume J VP = J with J n P; since J q P, there exists 1 1 r g V _ J such that ry J q P. Hence J n P n ry J and rJ n rP n J. Now, r has to be an element of P otherwise rP s P and we would get P n J. Ž y1 . 22 2 Thus JVPPPPPs rr JV srV . Hence JVsrV and by hypothesis rVP 2 WJ; thus r VPPPW JV s rV which can be possible only if r f P, contradiction.

2 Consider an ideal I of R with I n I; recalling the representation of I:I2 given by Eq.Ž. b we can state the following lemma which is, in some sense, a generalization of Proposition 2.7.

LEMMA 2.10. Let I be a nonidempotent ideal of R such that I is regular in Ž. 2 SR.If I RMMn IR for a maximal ideal M, then: Ž. Ž . 1 CI RGŽM.. Ž. Ž . 2 CIRGŽ M . sRP for a prime ideal P of R such that I P. Ž. 2 Ž. Proof. 1 We first claim that IRGŽ M . : I C IRGŽ M .. As recalled in 22Ž. Remark 1, IRMGs IR ŽM.; hence IRGŽM.nIRGŽM.. Now I : I : C I 22 2 and I s IIŽ:I.implies I : I CŽ.I and thus the claim follows. If it Ž. Ž. were C I : RGŽ M ., then clearly C IRGŽ M . sRGŽ M . and thus we would 2 have IRGŽ M . : I RGŽ M ., a contradiction,

Ž.2 By part Ž. 1 , C ŽIR . GŽ M . is a proper overring of the valuation domain RGŽ M .; hence it is of the form RP for a prime ideal P properly Ž. 2 Ž. 2 contained in GM. Thus I C IRGŽ M . sIRP and by the claim proved Ž. 2 Ž. in part 1 , IRPP=IR localizing at P . Lemma 2.9, applied to the valuation domain RGŽ M ., yields IRGŽ M . q PRGŽ M .; hence IRPPs R which means I P.

2 PROPOSITION 2.11. Let I be as in the preceding lemma. Assume I RM n Ž. Ž. IRM for e¨ery maximal ideal M containing I. Then C IIsCI. Ž. Proof. Let N be a maximal ideal of R such that N W I. Then C IIRN Ž. Ž. Ž. sCIRNM. Let M = I. Then C IIR sCIIRGŽM.sIRPPs R for a Ž. Ž. prime ideal of P as in Lemma 2.10. Thus C IIRGŽ M . sCIRM for every maximal ideal M of R; hence by Lemma 2.5, CŽ.IIsC Ž.I. In case the ideal I is contained only in a finite number of maximal ideals, then the necessary condition for the regularity of I expressed in Lemma 2.10 is also sufficient.

PROPOSITION 2.12. Let I be a nonidempotent ideal of R contained only in a finite number of maximal ideals. Then these are equi¨alent: Ž.1 I is regular in S Ž.R . CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 623

Ž. 2 Ž. 2 If M is a maximal ideal such that I RMMn IR , then C IRGŽM. sRP ,where P is a prime ideal of R and I P.

Proof. Ž.1 « Ž.2 . By Lemma 2.10. Ž. Ž. Ä4 2«1 . Let MjjgF be the finite set consisting of the maximal ideals of R containing I. Then I : I 2 ŽIR : IR2 .Ž.CI. s F jgFMjj Ml Let M , i F, be such that I 2 R IR . iMg iin M By Lemma 2.6, II2Ž:IR2.ŽIIR2Ž:IR2.. R MjijsFgFMMMjil I2CŽ.IR . We prove now that the two terms in the above intersection M i both contain IR and thus we conclude by Lemma 2.8. M i I2 CŽ.IR I2CŽ.IR IR2 R, by hypothesis; hence MGiis ŽM.s PPs I2CŽ.IR IR IR .Let j F.Thenwehave MGiii= ŽM.s M g IIR2Ž:IR2. R IR2Ž IR :IR2. R ; the last term coincides MMMMMMMjjijjjis with IR R IR since R is a valuation domain and thus IR is a MMji= M i Mj Mj regular element of S Ž.R . M j Before proving a sufficient condition on R in order for S Ž.R to be a Clifford semigroup, we prove a simple lemma.

LEMMA 2.13. Let R be a commutatië domain such that eëry nonzero element is contained only in a finite number of maximal ideals. Then, eëry ideal I of R contains a finitely generated ideal J, such that MŽ. R, I s MR Ž.,J

Proof. Let Ä4M1,...,Mn be the set of the maximal ideals containing I. Let 0 / x g I. Then x is contained in a finite number of maximal ideals, Ä4 say M1,...,Mnn,M q1,...,Mnqkj. For every j s 1,...,k let y g I _ Ž. Mnqj. Then J s x, y1,..., yk is clearly the wanted ideal.

THEOREM 2.14. Let R be a Prufer¨ domain such that e¨ery nonzero element is contained only in a finite number of maximal ideals. Then S Ž.Ris a Clifford semigroup.

Proof. Let I be an ideal of R. To prove that I is a regular element of SŽ.R, we apply Proposition 2.12; hence consider a maximal ideal M = I 2 with IRMGs IR ŽM.q I RGŽM.and let J be a finitely generated ideal contained in I with MRŽ.,J sMR Ž.,I.Jis invertible, thus J is regular 2 Ž.࠻ and clearly JRMMis principal over R ; hence JRMq JRMand JRMs Ž. MRMM. By Proposition 2.12, C JR sRPfor a prime ideal P W J. Thus Ž. Ž Ž. . CIRGŽ M . sCJRMG RŽM.sRRPGŽM.for P W I. Now we must have PnGMŽ.since P and GMŽ.are primes contained in M and I : GMŽ.. Ž. Thus C IRGŽ M . sRP with P W I.

COROLLARY 2.15. Let R be a Prufer¨ domain with a finite number of maximal ideals. Then S Ž.R is a Clifford semigroup. 624 S. BAZZONI

3. THE IDEMPOTENT ASSOCIATED TO A REGULAR ELEMENT OF SRŽ.

Rwill still denote a Prufer¨ domain. For every ideal I of R, D s I : I is described by Eq.Ž. a , namely

D R C Ž.I . s ž/F GŽM. l C MgMRŽ.,I

2 If I is a regular element of S Ž.R and L s II Ž:I .sID Ž:I ., then L is the idempotent associated to I Ž.see 2.1 . In order to characterize the ideals I such that I is regular, we investigate the properties of the D-ideal L.

LEMMA 3.1. Let I be a nonidempotent ideal of R and L s IDŽ.:I.The following hold: Ž. Ž . 1 If N f MR,I,then LRNNs R . Ž. Ž . 2 If M g MR,I then LRMGs LR ŽM.: RGŽM.. Ž. Ž. Ž. Proof. 1 LRNNs IR D : IR NNsRD:Iand thus LRNN= R , Ž. since D : I = R; on the other hand LRNNN: DR s R , by Eq. a . Ž. Ž . 2 Let M = I; recalling that IRMGs IR ŽM.see Remark 1 we have: Ž. Ž. LRMMs IR D : IR MGsIRŽM. D : IRMMG, but since RRŽM.sRGŽM. Ž. we obtain LRMGs IRŽM. D : IRGŽM.sLRGŽM.. Now L : D implies LRGŽ M . : DRGŽ M . which is clearly RGŽ M . since D : RGŽ M .. We find it convenient to split the set MRŽ.,I into three parts, namely:

M1s Ä4M g MRŽ.,IIR< MGprincipal over R ŽM.;

2 M2sÄ4MgMRŽ.,IIR

2 M3sÄ4MM<=I,IRMMs IR .

Ž. Remark 2. Recall that for every M g MR,I, IRGŽ M . is an archimedean ideal of RGŽ M .. Ž. If M g M2, then by Lemma 4.8 inwx 6 , IRGŽM. G M RGŽM.s IRGŽM.;if 2 MgM3, then, by Theorem 17.1 inwx 7 , IRMMs IR implies that IR Mis a Ž. prime ideal of RMMG, and hence IR s GMR ŽM. We can also consider:

A IR : I 2 R , where i 1,2,3. iGs FŽ.ŽM.GŽM. s MgMi CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 625

Remark 3. Notice that A R and 3 s F M g MG3 ŽM.

3 I:I2 ACŽ.I. sFilC is1

LEMMA 3.2. Let I, L be as in Lemma 3.1 and assume M g MRŽ.,I. Then: Ž. 1 If LRGŽ M . s RGŽ M ., then IRGŽ M . is principal o¨er RGŽ M .. Ž. Ž. 2 If M g M3, then LRGŽM.s GMRGŽM.. Ž. Ž. 3 If I is a regular element of S R and M g M1, then LRMGs R ŽM.. Ž. Ž. 4 If I is a regular element of S R and M g M2 , then LRM s Ž. LRGŽM.s GMRGŽM.. Ž. Proof. 1 Let M = I be such that LRGŽ M . s RGŽ M .. Write D s RGŽ M . Ä Ž. 4 Ž. lD11where D s F RMGŽMЈ.< ЈgMR,I,MЈ/M lCI. Then Ž.Ž. D:IsRG Ž M . :IR G ŽM . l D 1 : I ;thus LRG ŽM . s Ž.Ž. IRGŽ M . RGŽ M . : IRGŽ M . l IRGŽ M . D1 : I . Now, by Lemma 1 inwx 2 , IRGŽ M .Ž. RGŽ M . : IRGŽ M . is either RGŽ M . or GMR Ž.GŽ M . according to I principal or not over RGŽ M ..If IRGŽ M . were not principal over RGŽ M . we Ž. Ž. would get RGŽ M . s LRGŽ M . s GMRGŽ M . lIRGŽ M . D1 : I : GMRŽ.GŽ M ., a contradiction. Ž. 2IfMgM3, then by the Remark 2 we obtain LRGŽ M . s Ž. Ž.Ž. Ž. IRGŽM. D : IRGŽM.sGMRGŽM. D:I. Hence LRGŽM.= GMRGŽM., Ž. Ž. but LRGŽM.cannot be RGŽM.by part 1 and thus LRGŽM.s GMRGŽM.. Ž. 3 Notice that the hypothesis I regular implies IRGŽ M . LRGŽ M . s Ž. IRGŽ M . for every M g MR,I. Now if M g M1, then IRGŽ M . s aRGŽ M . for an element a g M; hence aLRGŽ M . s aRGŽ M . implies LRGŽ M . s RGŽ M .. Ž. 4 Let M g M2 . IRGŽ M . LRGŽ M . s IRGŽ M . implies LRGŽ M . = Ž. Ž. Ž .࠻ GMRGŽ M .. In fact, GMRGŽ M . s IRGŽ M . and if it were LRGŽ M . n Ž. GMRGŽ M . we would get IRGŽ M . LRGŽ M . n IRGŽ M . by the definition of the prime associated to an ideal in a valuation domain. But LRGŽ M . Ž. cannot be RGŽ M ., by part 1 of this lemma; hence we conclude LRGŽ M . s GMRŽ.GŽM.. PROPOSITION 3.3. Let I be a nonidempotent ideal of R. These are equi¨a- lent:

Ž.1 Iisinërtible oër D s I : I. Ž. Ž. 2 I is regular in S R and IRMG is principal oër RŽM. for eëry MgMRŽ.,I. 626 S. BAZZONI

Proof. Ž.1 « Ž.2. I is regular by Lemma 1.3. For every maximal ideal Ž. Ž. M=I,D:RGŽ M ., by Eq. a ; hence by Theorem 26.1 inwx 7 , GMDis a proper prime ideal of D and DGŽ M . DGs R ŽM.. Moreover I is finitely generated over D, and thus IDGŽ M . DGis principal over D ŽM.DGs R ŽM. and IRGŽ M . s IRM . Ž.2« Ž.1 . We prove that L s IDŽ.:I is equal to D. Let N W I. Then Ž. by Lemma 3.1 1 , LRNNs R ;if M=I, then by hypothesis M belongs to Ž. M1; hence by Lemma 3.2 3 , LRMGs R ŽM.. Thus, by Lemma 2.5, L s R RD. FM=IGŽM.lFNWINs

COROLLARY 3.4. If I is a regular element of S Ž.R and IRM is principal Ž. o¨er RM for e¨ery maximal ideal M g Max R , then I is a finitely generated ideal of R.

Proof. In these hypothesis I : I s R, by Eq.Ž. a . Hence the conclusion follows by Proposition 3.3. We end this section by giving two characterizations of the regular elements I of S Ž.R , one in terms of the idempotent associated to I; the other based on properties of the ideal I.

PROPOSITION 3.5. Let I be a nonidempotent ideal of R. I is a regular element of S Ž.R if and only if the following conditions hold. Ž. 1 For e¨ery M g M1, LRMGs R ŽM.. Ž. Ž. 2 For e¨ery M g M23j M , LRMGs GMR ŽM.. Hence L R GMRŽ.C Ž.I. s F MgMG12ŽM.l F MgMjMG3ŽM.l Proof. The necessity is given by Lemma 3.2. Ž. Ž. For the sufficiency notice that 1 and 2 imply ILRMMs IR for every Ž. Ž. M=Iby the Remark 2 . If N W I, then by Lemma 3.1 1 , ILRNNs IR s Ž. RN . Hence by Lemma 2.5 IL s I and thus I is a regular element of S R . The last statement is clear. In the notations introduced above we have:

PROPOSITION 3.6. Let I be a nonidempotent ideal of R. I is a regular Ž. Ž . 2 element of S R if and only if for e¨ery M g MR,I with IRMMq I R the following hold: Ž. 2 1 IARiM=IR M for i s 1, 2, 3. Ž. Ž . 2 CIRGŽ M . sRP for a prime ideal P W I. Proof. Assume I is regular. ThenŽ. 1 follows by Lemma 2.8 and Remark 3;Ž. 2 follows by Lemma 2.10. CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 627

Assume now thatŽ. 1 and Ž. 2 hold. To prove that I is regular we apply 22Ž. Lemma 2.8. Let M = I with IRMMnIR .Then ICIRMs 2Ž. Ž. Ž. I CIRGŽ M . sRP by 2 ; hence the conclusion follows by 1 and Re- mark 3.

4. DOMAINS SATISFYING PROPERTY Ž.)

We say that a commutative domain R satisfies propertyŽ. * if the following holds:

Ž.* An ideal I of R is invertible if and only if every localization IRM at a maximal ideal M is invertible. For a Prufer¨ domain R propertyŽ. * is equivalent to: Ž.* An ideal I of R is finitely generated if and only if every localization IRM at a maximal ideal M is principal. Clearly the above condition can be equivalently formulated for fractional ideals of R. Notice that an almost Dedekind domain not Dedekind does not satisfy propertyŽ. * . In view of Corollary 3.4, the class of the Prufer¨ domains such that S Ž.R is a Clifford semigroup is contained in the class of Prufer¨ domains with propertyŽ. * . In this section we will consider the Prufer¨ domains satisfying property Ž.*. Remark 4. Let M and N be two distinct maximal ideals of the Prufer¨ domain R. Then the set of the prime ideals contained in M l N is a chain whose union P is the largest prime ideal contained in M l N; this P will be denoted by M n N. Moreover RRMNsR P. Denote by T the subset of MaxŽ.R consisting of the maximal ideals M such that R Ä RM< MaxŽ.R , M / M4. MMiWF ig i The class of Prufer¨ domains such that T s MaxŽ.R has been studied in wx8᎐10 . The Prufer¨ domains in this class are said to satisfy property Ž.࠻ .In particular, in Proposition 1.4wx 10 , it is proved that M g T if and only if there exists a finitely generated ideal I of R such that the only maximal ideal containing I is M. LEMMA 4.1. Let R be a Prufer¨ domain satisfying property Ž.* and let M be a maximal ideal of R. The following are equi¨alent: Ž.1 R ÄRM< MaxŽ.R , M / M4. MiWF gIMi ig i Ž. 2 MqFig IiP where P i is the largest prime contained in M l Mi Ž. for e¨ery Miig Max R , M / M. 628 S. BAZZONI

Proof. Ž.1 « Ž.2 . Let I be a finitely generated ideal contained in M but not contained in any other maximal ideal Mi. Then, by Theorem 4.10 inwx 7 , I s IRMMl R; IR is principal and thus it coincides with aRMfor an element a M. Now for every M MaxŽ.R , M / M, we have: IR g iig Mi RaR R R aR R ; hence aR R . We claim that sMMMMPMiiiiiiisl s l PM= aPfor every i. In fact, R is a valuation domain and thus, if a P , fiMi g i then aR PR PR where the last equality holds by Remark 1. PiPiMii: s i Thus a fDiiP. Ž. Ž. 2«1 . Choose an element a g MrD iiP and let I s aRMl R. Localizing I at M we obtain IR aR R R R R ; i Mis PMPMM iiiiil s l s hence the only maximal ideal containing I is M. Clearly IRMMs aR and since R satisfies propertyŽ. * we conclude that I is a finitely generated ideal of R. The next proposition is crucial in determining a wide class of Prufer¨ domains satisfyingŽ. * . PROPOSITION 4.2. Let R be a Prufer¨ domain satisfying Ž.*.Then e¨ery nonzero element of R is contained only in a finite number of maximal ideals belonging to T. Ž. Ž. Ä␣ Proof. Let 0 / x g F ␣ gnM␣␣, M g Max R and let T x s g 4 ⌳

Let M be an arbitrary maximal ideal of R.Then BR M s Ý y1 ␣ Ž. ␣ g TŽ x. AR␣ M . Now, A␣ is finitely generated for every g T x , and y1 hence we have: A␣ RMMs R : AR␣Mand AR␣MMis R unless M s M␣ in which case ARM␣ ␣ is principal. Hence, BRMMis either R or, if ␣ TŽ.x, then BR Ž.AR y1 is principal. Thus, as claimed B is g M␣ s ␣ M␣ finitely generated, namely n 1 B A y s ÝŽ.␣i is1 Ä4Ž. Ž.Ž .y1 for a finite subset ␣1,...,␣n of T x . Now, for every ␣ g T x , A␣ : B and thus we have n By1 A A M s F ␣ i: ␣␣: is1 CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 629

Ž.Ž. since the A␣’s and B are finitely generated . But if ␣ g T x , ␣ / ␣i for every i 1,...,n, then BRy1 n Ž.A,R R; hence T Ž.x s Mi␣sFs1␣iMM␣␣s s Ä4M,...,M . ␣1 ␣n

THEOREM 4.3. Let R be a Prufer¨ domain satisfying one of the following two conditions:

Ž.1 RM is finite-dimensional for each maximal ideal M of R. Ž.2 R has property Ž.࠻ .

Then R satisfies property Ž.* if and only if e¨ery nonzero element of R is contained only in a finite number of maximal ideals.

Proof. The sufficient condition follows by Lemma 37.3 inwx 7 . We prove now the necessary condition for a Prufer¨ domain satisfying Ž.1 . Let M be a maximal ideal of R; M has finite dimension and thus

M qFig IiP where Piis M n Mifor every M / Mi. Hence, by Lemma 4.1, M g T and thus the conclusion follows by Proposition 4.2. If R has property Ž.࠻ then the conclusion follows by Proposition 4.2.

Remark 5. ConditionŽ. 1 Theorem 4.3 replaces the condition ‘‘R has finite dimension’’ which appeared in the first version of the paper. We thank the referee for pointing out that our proof applies to the larger class of Prufer¨ domains with finite-dimensional localizations at maximal ideals. We will show, by the next example, that the class of Prufer¨ domains satisfying conditionŽ. 1 is strictly larger than the class of finite-dimensional Prufer¨ domains.

EXAMPLE C. We give an example of an infinite-dimensional Bezout domain R such that every localization of R at a maximal ideal is finite- dimensional. In view of Theorem 18.6 and Proposition 19.11 inwx 7 it suffices to construct a lattice ordered abelian group G such that G admits totally ordered factor groups of any finite rank, but it does not admit totally ordered factor groups of infinite rank.

For every n g ގ, let Tn be the direct sum of n copies of the group of integers with the lexicographic order and define G T as the s [ng ގ n weak cardinal sum of the totally ordered groups Tn Žthe order is defined . componentwise . For every n g ގ, Tn is an ordered factor group of G of rank n.If His a subgroup of G such that GrH is totally ordered, then H must be a convex sublattice of G with the property that, for each element g g G either gq or gy belongs to H. With these observations it is not hard to prove that, if GrH is totally ordered, there exists an index n0 g ގ such that H contains the subgroup T ; hence G H is isomorphic to [n/ n0 n r a subgroup of T . n0 630 S. BAZZONI

We do not have an example of a Prufer¨ domain satisfying propertyŽ. * and not satisfying Ž.࠻ , but if there is one then it must satisfy the following condition.

PROPOSITION 4.4. Let R be a Prufer¨ domain satisfying Ž.* but not Ž࠻ .. Then the set MaxŽ.R _ T is infinite. Proof. Let M be a maximal ideal of R such that M f T. Then, by Lemma 4.1, M s D ig IiP where Piis the maximum prime contained in Ä 4 MlMii. Notice that the set Pi

Ž.Ј If P is a prime ideal and P n Q, there exists a finitely generated ideal I such that P : I : Q. A slightly different formulation of property Ž.Ј was introduced inwx 4 ; there it is proved that R satisfies Ž.Ј for each pair of prime ideals P and Q,if and only if every prime ideal P of R is a maximal ideal of the overring P : P.

PROPOSITION 4.5. Let R be a Prufer¨ domain satisfying property Ž.*.Then R satisfies the separation property Ž.Ј .

Proof. Let P n Q and take x g Q _ P. We claim that I s P q Rx is finitely generated from which we will get the conclusion. Since R satisfies CLASS SEMIGROUPS OF PRUFER¨ DOMAINS 631

Ž.* , it is enough to show that every localization of I at a maximal ideal M containing I is principal. Now IRMMMMs PR q xR s xR where the last equality holds since P is a prime ideal. We can now summarize the results obtained up to now in the following theorem.

THEOREM 4.6. Let R be a Prufer¨ domain satisfying one of the following two conditions:

Ž.1 RM is finite-dimensional for each maximal ideal M of R. Ž.2 R has property Ž.࠻ . Then the class semigroup S Ž.R of R is a Clifford semigroup if and only if e¨ery nonzero element of R is contained only in a finite number of maximal ideals. Proof. The statement follows by applying Theorem 2.14, Corollary 3.4, and Theorem 4.3. Notice that if the conjecture were true, then we would have that for a Prufer¨ domain R, S Ž.R is a Clifford semigroup if and only if every nonzero element of R is contained only in a finite number of maximal ideals.

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