JOURNAL OF ALGEBRA 184, 613]631Ž. 1996 ARTICLE NO. 0279 Class Semigroups of PruferÈ Domains S. Bazzoni* Dipartimento di Matematica Pura e Applicata, Uni¨ersita di Pado¨a, ¨ia 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 PruferÈ 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.a property, introduced in 1967 for PruferÈ domains R, stating that if D12and D are two distinct sets of maximal ideals of R, Ä 4Ä 4 then F RMM <gD1 /FRMM <gD2 . Let C be the class of PruferÈ domains satisfying the separation property Ž.a or with the property that each localization at a maximal ideal if finite-dimensional. We prove that, if R belongs to C, then the local invertibility property holds on R if and only if every nonzero element of R is contained only in a finite number of maximal ideals of R. Moreover if R belongs to C, then 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 of R. Q 1996 Academic Press, Inc. 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 multiplica- tion 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 Q 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 PruferÈ domain. The aim of this paper is to solve the problem, raised inwx 2 , namely to give a characterization of the class of the PruferÈ 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 PruferÈ 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 PruferÈ 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 PruferÈ domains satisfying propertyŽ. * :. Clearly a PruferÈ domain has propertyŽ. * if and only if every ideal that is locally invertible at each maximal ideal is finitely generated. A PruferÈ 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 PRUFERÈ 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.a property defined inwx 9, 10 as follows: the PruferÈ domain R satisfies the Ž.a property if and only if for every two distinct sets D12and D consisting of maximal ideals of R, Ä 4Ä 4 Ž. FRMM <gD1 /FRMM <gD2 . Inwx 10 it is shown that the a prop- erty holds if and only if, for every maximal ideal M, there exists a finitely generated ideal I contained in M but not contained in any maximal ideal different from M. We do not have an example of a PruferÈ domain satisfying propertyŽ. * and not satisfying Ž.a , but we prove that if such an R exists then it must have infinitely many maximal ideal M's such that M s D Piiwhere P denotes the largest prime ideal contained in M l Mi for every maximal ideal Mi distinct from M. Moreover we prove that a PruferÈ domain satisfying propertyŽ. * has the following separation property: 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 PruferÈ 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 PruferÈ 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.a . 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: fef: 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 isomor- phism classes of nonzero fractional ideals of R.