A characterization of large Dedekind domains

Abstract

Let D be a commutative domain with identity, and let $${\mathcal {L}}(D)$$ be the lattice of nonzero ideals of D. Say that D is ideal upper finite provided $${\mathcal {L}}(D)$$ is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D. Now let $$\kappa >2^{\aleph _0}$$ be a cardinal. We show that a domain D of cardinality $$\kappa $$ is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if $$\kappa $$ is a cardinal such that $$\aleph _0\le \kappa \le 2^{\aleph _0}$$ , then there is an ideal upper finite domain of cardinality $$\kappa $$ which is not Dedekind.