Maximal non-decreasing subring of its quotient field

Abstract

An integral domain R is called maximal non-going down subring of its quotient field, if R is not going down, and every proper overring of R is going down. We do prove that R is a maximal non-going down subring of its quotient field if and only if R  is a quasi-local domain with maximal ideal m and its integral closure R is a semi-local Prufer domain with two maximal ideals M, N such that M intersect N = m, the extension R is not going down, and R is the unique quasi-local subring of R.