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

DOI:
https://doi.org/10.1090/S0002-9939-02-06816-8

Published electronically:
November 14, 2002

MathSciNet review:
1974630

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an integral domain with quotient field and integral closure . An overring of is a subring of containing , and denotes the set of overrings of . We consider primarily two finiteness conditions on : (FO), which states that is finite, and (FC), the condition that each chain of distinct elements of is finite. (FO) is strictly stronger than (FC), but if , each of (FO) and (FC) is equivalent to the condition that is a Prüfer domain with finite prime spectrum. In general satisfies (FC) iff satisfies (FC) and all chains of subrings of containing 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

DOI:
https://doi.org/10.1090/S0002-9939-02-06816-8

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