A special chain theorem in the set of intermediate rings

Abstract

Let [Formula: see text] be an extension of integral domains, and let [Formula: see text] be the integral closure of [Formula: see text] in [Formula: see text]. The main purpose of this paper is to study [Formula: see text], the set of intermediate rings between [Formula: see text] and [Formula: see text]. As a main tool, we establish an explicit description of any intermediate ring in terms of localizations of [Formula: see text] (or [Formula: see text]). This study effectively enables us to characterize the minimal extensions in [Formula: see text]. We also prove a special chain theorem concerning the length of an arbitrary maximal chain in [Formula: see text].