Abstract
We determine the Dedekind domain pairs of rings; that is, pairs of rings R ⊂ S such that each intermediary ring in between R and S is a Dedekind domain. We also establish that if R ⊂ 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 |Supp(S/R)| < ∞ or |Max(R)| < ∞, (3) S is a field, (4) R is a valuation domain, and (5) R ⊂ S is an integral extension. We also provide some classifications of pairs of rings having exactly two non‐Dedekind domain intermediary rings.
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