On a class of rings that generalize Dedekind rings
Let \(A\) be a commutative integral domain and let \({\mathcal F}\) be a localizing system of ideals of \(A\). One can then construct a localization \(A_{\mathcal F}\) of \(A\), which is a subring of the field of fractions \(K\) of \(A\). The ring \(A\) is said to be a generalized Dedekind domain if conversely any ring \(B\) with \(A \subseteq B \subseteq K\) can be realized in the form \(A_{\mathcal F}\) for some such \({\mathcal F}\). In this paper, the authors first give a characterization of Prüfer rings in terms of their localizations, followed by many alternative descriptions of generalized Dedekind domains. These in turn lead to a number of characterizations of standard Dedekind domains. They then consider extensions of a generalized Dedekind domain in an algebraic extension of \(K\), not necessarily finite, and give criteria for such an extension to be itself generalized Dedekind. The results are illustrated with a number of explicit examples.
scientific article; zbMATH DE number 763934