On the finiteness conditions of the set of intermediate rings

This book comes within the scope of Commutative Algebra and studies problems related to the finiteness conditions of the set of intermediate rings. Let S be a ring extension of R and R the integral closure of R in S. We first characterize minimal extensions and give a special chain theorem concerning the length of an arbitrary maximal chain of rings in [R,S], the set of intermediate rings. As the main tool, we establish an explicit description of any intermediate ring in terms of localization of R (or R ). In a second part, we are interested to study the behavior of [R,S]. Precisely, we establish several necessary and sufficient conditions for which every ring contained between R and S compares with R under inclusion. This study 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]. Our final contributions are the FIP extensions. This kind of extensions was considered to generalize the Primitive Element Theorem. We give a complete generalization of the last cited theorem in the context of an arbitrary ring extension.