Some finiteness conditions on the set of overrings of an integral domain
Author:
Robert Gilmer
Journal:
Proc. Amer. Math. Soc. 131 (2003), 23372346
MSC (2000):
Primary 13G05, 13B02, 13B22, 13F05
Published electronically:
November 14, 2002
MathSciNet review:
1974630
Fulltext PDF Free Access
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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 323064510
Email:
gilmer@math.fsu.edu
DOI:
http://dx.doi.org/10.1090/S0002993902068168
PII:
S 00029939(02)068168
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
