Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings - Naseam Al-Kuleab; Noômen Jarboui (Journal of Mathematics)
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 SuppS/R<∞ or MaxR<∞, (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.