Maximal non-treed subring of its quotient field

Abstract

An integral domain R is called maximal non-treed subring of its quotient field, if R is not treed, and every proper overring of R is treed. We do prove that R is a maximal non-treed subring of its quotient field if and only if R is non-treed, local with maximal ideal m and its integral closure \(\overline{R}\) is a semi-local Prufer domain with two maximal ideals M, N such that \(M\cap N=m\), and there is no field lying properly between R / m and \(\overline{R}/M\times \overline{R}/N\). Additional characterizations are settled in terms of pullback rings or minimal overrings.