On Matlis domains and Prüfer sections of Noetherian domains

The class of Matlis domain, those integral domains whose quotient field has projective dimension 1, is surprisingly broad. However, whether every domain of Krull dimension 1 is a Matlis domain does not appear to have been resolved in the literature. In this note we construct a class of examples of one-dimensional domains (in fact, almost Dedekind domains) that are overrings of K[X, Y ] but are not Matlis domains. These examples fit into a larger context of what we term “Prüfer sections” of Noetherian domains, a notion we also consider briefly, and with emphasis on Prüfer sections of two-dimensional Noetherian domains.