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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Some finiteness conditions on the set of overrings of an integral domain
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by Robert Gilmer PDF
Proc. Amer. Math. Soc. 131 (2003), 2337-2346 Request permission

Abstract:

Let $D$ be an integral domain with quotient field $K$ and integral closure $\overline D$. An overring of $D$ is a subring of $K$ containing $D$, and $\mathcal {O}(D)$ denotes the set of overrings of $D$. We consider primarily two finiteness conditions on $\mathcal {O}(D)$: (FO), which states that $\mathcal {O}(D)$ is finite, and (FC), the condition that each chain of distinct elements of $\mathcal {O}(D)$ is finite. (FO) is strictly stronger than (FC), but if $D=\overline {D}$, each of (FO) and (FC) is equivalent to the condition that $D$ is a Prüfer domain with finite prime spectrum. In general $D$ satisfies (FC) iff $\overline {D}$ satisfies (FC) and all chains of subrings of $\overline {D}$ containing $D$ have finite length. The corresponding statement for (FO) is also valid.
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Additional Information
  • Robert Gilmer
  • Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510
  • Email: gilmer@math.fsu.edu
  • Received by editor(s): January 15, 2002
  • Received by editor(s) in revised form: March 27, 2002
  • Published electronically: November 14, 2002
  • Communicated by: Wolmer V. Vasconcelos
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 2337-2346
  • MSC (2000): Primary 13G05, 13B02, 13B22, 13F05
  • DOI: https://doi.org/10.1090/S0002-9939-02-06816-8
  • MathSciNet review: 1974630