Prüfer Domains with Clifford Class Semigroup
چکیده
Bazzoni’s Conjecture states that the Prüfer domain R has finite character if and only if R has the property that an ideal of R is finitely generated if and only if it is locally principal. In [4] the authors use the language and results from the theory of lattice-ordered groups to show that the conjecture is true. In this article we supply a purely ring theoretic proof. 1. Bazzoni’s Conjecture Throughout all integral domains are assumed to be commutative. For an integral domain R, F (R) denotes the semigroup of fractional ideals of R (under ideal multiplication) while P(R) denotes the subsemigroup consisting of principal ideals. The class semigroup of R is the factor semigroup F (R)/P(R) and is denoted S (R). A semigroup S is called a Clifford semigroup when every element is regular in the sense of von Neumann, that is, for every a ∈ S there is an s ∈ S for which as = a. The domain R is called a Clifford regular domain when S (R) is Clifford regular. In the article [1] S. Bazzoni proved that if a Prüfer domain has finite character (that is, every nonzero element belongs to a finite number of maximal ideals) then S (R) is a Clifford semigroup, and in turn, if S (R) is a Clifford semigroup, then R satisfies (∗) (defined below). In a later article, [2], she was able to show that if S (R) is a Clifford semigroup, then R has finite character. In [1] and then again in [2] she proposed the following Conjecture: A Prüfer domain satisfies property (*) if and only if R has finite character. Recently, the authors of [4] proved using techniques from the theory of latticeordered groups that the conjecture is indeed true. The main road used in their proof was to translate the concepts discussed above into the language of `-groups via the lattice-ordered group of invertible ideals of a Prüfer domain. Once the translations were made they used several old and well-known results to finish the proof. In this article we give a purely ring-theoretic proof of the validity of Bazzoni’s conjecture. Our proof is mainly a translation from `-groups to ring theory of the proof from [4] except in one crucial place. We elaborate on this matter. Given a Prüfer domain R and G its `-group of invertible ideals any information about an `-homomorphic image of G can be translated to information about an appropriate localization of R. On the other hand, and unfortunately, there is no known ring-theoretic construction that allows one to gather information about the 2000 Mathematics Subject Classification. Primary 13F05.