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

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150]. [ABSTRACT FROM PUBLISHER]