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Maximal non-integrally closed subrings of an integral domain

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Abstract

Let \(R\subset S\) be an extension of integral domains. The domain R is said to be a maximal non-integrally closed subring of S if R is not integrally closed in S, while each subring of S properly containing R is integrally closed in S. Jaballah (J Algebra Appl 11(5):1250041, 18pp, 2012) has characterized these domains when S is the quotient field of R. The main purpose of this paper is to study this kind of ring extensions in the general case. Some examples are provided to illustrate our obtained results. Our main result also answers a key question raised by Gilmer and Heinzer (J Math Kyoto Univ 7(2):133–150, 1967).

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Acknowledgements

The authors extend their thanks to the referee for his/her valuable comments and suggestions which helped to improve the presentation of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia

    Noômen Jarboui & Suaad Aljubran

  2. Département de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, Route de Soukra, P.O. Box 1171, Sfax, 3038, Tunisia

    Noômen Jarboui

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  1. Noômen Jarboui
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  2. Suaad Aljubran
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Correspondence to Noômen Jarboui.

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Jarboui, N., Aljubran, S. Maximal non-integrally closed subrings of an integral domain. Ricerche mat 71, 325–332 (2022). https://doi.org/10.1007/s11587-020-00500-0

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