A characterization of large Dedekind domains
A commutative domain \(D\) (with identity) is said to be \textit{ideal upper finite} if every nonzero ideal of \(D\) is contained in but finitely many ideals of \(D\). The author proves that being a Dedekind domain is a sufficient but not necessary condition for being an ideal upper finite domain. However, if \(D\) has cardinality \(\kappa >2^{\aleph_0}\), \(D\) is a Dedekind domain if and only if \(D\) is ideal upper finite. In ZFC, the cardinal \(2^{\aleph_0}\) is the best possible, since if \(\kappa\) is a cardinal \(\aleph_0 \leq \kappa \leq 2^{\aleph_0}\), there exists an ideal upper finite domain \(D\) of cardinality \(\kappa\) which is not Dedekind.
scientific article; zbMATH DE number 7217591