Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings

Abstract

We determine the Dedekind domain pairs of rings; that is, pairs of rings $R\subset S$ such that each intermediary ring in between $R$ and $S$ is a Dedekind domain. We also establish that if $R\subset S$ is an extension of rings having only one non-Dedekind intermediary ring, then necessarily $R$ is not Dedekind and so $R$ is a maximal non-Dedekind domain subring of $S$ . Maximal non-Dedekind domain subrings $R$ of $S$ are identified in the following cases: (1) $R$ is not integrally closed, (2) $R$ is integrally closed and either $\left|\text{Supp}\left(S/R\right)\right|<\infty$ or $\left|\text{Max}\left(R\right)\right|<\infty$ , (3) $S$ is a field, (4) $R$ is a valuation domain, and (5) $R\subset S$ is an integral extension. We also provide some classifications of pairs of rings having exactly two non-Dedekind domain intermediary rings.