Some factorization properties of Krull domains with infinite cyclic divisor class group

In this paper, we study factorization properties of Krull domains with divisor class group . This continues a preliminary study of Dedekind domains with class group in Section IV of [7]. In section 1, using the Φ-function we introduce the notion of a Φ-finite domain and then determine the relationship between these domains and BFDs and RBFDs (see [1]). In particular, we show that a Φ-finite domain need not be an RBFD. In Section 2, we obtain necessary and sufficient conditions on the set S of divisor classes of D which contain height-one prime ideals so that D is Φ-finite. This leads to the following result: if D is a Krull domain with divisor class group , then D is Φ-finite if and only if D is an RBFD. We also find a bound for the elasticity, ϱ(D), of the domain D and show in Section 3 that, unlike the case where the divisor class group of D is finite, the elasticity of D may not be “attained” by the factorization of a single element