Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13G05, 13B02, 13B22, 13F05

Retrieve articles in all journals with MSC (2000): 13G05, 13B02, 13B22, 13F05


Additional Information

Robert Gilmer
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510
Email: gilmer@math.fsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06816-8
PII: S 0002-9939(02)06816-8
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