It follows that both 5 and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x] s Q R[x] n . The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially interesting paper concerning R(x) is that of Arnold's [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work. In § 2 we determine conditions on the

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... g R so that R(x) = R[x] v . A complete characterization of this property is given for Noetherian domains in Proposition 2.2. In particular, we prove that if R is a Noetherian domain, then R(x) = R[x] v if and only if depth (R) ^ 1. Some sufficient conditions for R(x) = R[x]u are that R be treed (Proposition 2.5), or that SP (R) (see § 2 for définitions) be finite (Proposition 2.9). The main results of this paper occur in § 3. We prove that if R is either a GCD-domain, an integrally closed coherent domain, or a Krull domain, then R[x]u is a Bezout domain. As is well known [7, Theorem 32.7], the Kronecker function ring R K of an integrally closed domain R is a Bezout domain. Hence, it would seem likely that R K and R[x] v would coincide for many rings R. However, this is not the case. In fact, R K = R[x] v if and only if R is a Priifer domain. In general there is no containment relation between R K and R[x] v (Remark 3.3). Finally, we apply the results of § 3 to § 1 to obtain new characterizations of Priifer domains, Bezout domains, and Dedekind domains (Corollary 3.2). When does Ut it is natural to consider when this inclusion is strict or not. We shall indicate some classes of domains establishing both possibilities. In Section 3, we will further pursue this topic as an application of the results of that section.