Abstract
The notion of the unique maximal overring of an integral domain is introduced and the domains for which the integral closure is the unique maximal overring are characterized. We characterize ring extensions \(J\subseteq S\), satisfying a certain technical condition, for which \(|\text {MSupp}_J(S/J)|=1.\) The applications of the preceding result include, an extension of Ben Nasr and Zeidi’s result (Nasr and Zeidi 2017, Theorem 2.7) (Ben Nasr and Zeidi, When is the integral closure comparable to all intermediate rings, Bulletin of Australian Mathematical Society, 95 (2017), 14–21) for a ring extension \(R\subseteq S\) to be pinched at the integral closure \(\overline{R}\) of R in S and an extension of a result by Dobbs and Jarboui (Dobbs and Noomen 2022, Theorem 3.9) (D.E. Dobbs and Noomen Jarboui, Prüfer-closed extensions and FCP \(\lambda \)-extensions of commutative rings, Palestine Journal of Mathematics, Vol. 11(3) (2022), 362–378.). We generalize the main result of Gilbert ( (6, Proposition 2.8, Chapter II), Extensions of Commutative Rings With Linearly Ordered Intermediate Rings, Doctoral Dissertation, University of Tennessee-Knoxville). This result plays a key role in Gilbert’s alternative proof of Ferrand and J Olivier’s characterization of minimal ring extensions. We are also able to partially characterize the local rings which are not Prüfer closed and have the unique minimal overring without assuming any finiteness hypothesis.
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Acknowledgements
The first author was supported by Junior Research Fellowship from UGC, India. The second author was supported by R &D grant, NBHM-DAE, India. We also take this opportunity to thank the anonymous reviewers for their invaluable comments, corrections, and suggestions. Their meticulous feedback significantly enhanced the clarity and coherence of the paper, ensuring its quality and impact. In particular, Lemma 2.7 owes its existence to the suggestion of one the reviewers, this in turn has resulted in the simplification of the proof of Theorem 3.2.
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Singh, M., Singh, R. On ring extensions pinched at the integral closure. Beitr Algebra Geom (2024). https://doi.org/10.1007/s13366-024-00752-z
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DOI: https://doi.org/10.1007/s13366-024-00752-z
Keywords
- Comparable overring
- Integral closure
- Valuation domain
- Pseudovaluation domain
- I-domain
- Prüfer hull
- Prüfer-closed