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Some finiteness conditions on the set of overrings of an integral domain

Author: Robert Gilmer
Journal: Proc. Amer. Math. Soc. 131 (2003), 2337-2346
MSC (2000): Primary 13G05, 13B02, 13B22, 13F05
Published electronically: November 14, 2002
MathSciNet review: 1974630
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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.

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Additional Information

Robert Gilmer
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510

Keywords: Integral domain, overring, finite chains of overrings, finite prime spectrum, Pr\"ufer domain
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
Article copyright: © Copyright 2002 American Mathematical Society

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