Some finiteness conditions on the set of overrings of an integral domain
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- by Robert Gilmer PDF
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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.References
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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