On the finiteness conditions of the set of intermediate rings
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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 establ...
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 consideredto generalize the Primitive Element Theorem. We give a complete generalization of the last cited theorem in the context of an arbitrary ring extension.
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