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WHEN IS THE INTEGRAL CLOSURE COMPARABLE TO ALL INTERMEDIATE RINGS

Part of: Ring extensions and related topics Chain conditions, finiteness conditions General commutative ring theory

Published online by Cambridge University Press:  19 October 2016

MABROUK BEN NASR  and
NABIL ZEIDI
MABROUK BEN NASR
Affiliation:
Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, B.P. 1171, 3000 Sfax, Tunisia email [email protected]
NABIL ZEIDI*
Affiliation:
Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, B.P. 1171, 3000 Sfax, Tunisia email [email protected]
*
✉ [email protected]
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Abstract

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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].

Keywords

intermediate ringminimal ring extensionfinite chain conditionmaximal chainnormal pairsupportconductor

MSC classification

Primary: 13B02: Extension theory
Secondary: 13A18: Valuations and their generalizations 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 13B25: Polynomials over commutative rings 13B35: Completion 13E05: Noetherian rings and modules
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 14 - 21
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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