pp 225238
How to Construct Huge Chains of Prime Ideals in Power Series Rings
 Byung Gyun KangAffiliated withDepartment of Mathematics, Pohang University of Science and Technology Email author
 , Phan Thanh ToanAffiliated withDepartment of Mathematics, Pohang University of Science and Technology
Abstract
Let R be a commutative ring with identity. It is well known that if each chain of prime ideals in R has length at most n, then each chain of prime ideals in the polynomial ring R[X] has length at most 2n + 1. For the power series ring R[[X]], there is however no similar upper bound on lengths of chains of its prime ideals. In fact, in some special cases, there may exist chains of prime ideals in R[[X]] with huge lengths (e.g., \(2^{\aleph _{1}}\)) even if each chain of prime ideals in R has length at most one. The purpose of this work is to give a brief review on known constructions of chains of prime ideals in R[[X]] in those cases. By taking into account the techniques which are used in the constructions and possibly by applying some new tools, we hope to construct huge chains of prime ideals in R[[X]] in more general cases.
Keywords
Almost Dedekind domain Krull dimension NonSFT ring Power series ring Valuation domainMathematics Subject Classification (2010):
13A15 13C15 13F05 13F25 13F30. Title
 How to Construct Huge Chains of Prime Ideals in Power Series Rings
 Book Title
 Commutative Algebra
 Book Subtitle
 Recent Advances in Commutative Rings, IntegerValued Polynomials, and Polynomial Functions
 Pages
 pp 225238
 Copyright
 2014
 DOI
 10.1007/9781493909254_13
 Print ISBN
 9781493909247
 Online ISBN
 9781493909254
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media New York
 Additional Links
 Topics
 Keywords

 Almost Dedekind domain
 Krull dimension
 NonSFT ring
 Power series ring
 Valuation domain
 13A15
 13C15
 13F05
 13F25
 13F30.
 eBook Packages
 Editors

 Marco Fontana ^{(1)}
 Sophie Frisch ^{(2)}
 Sarah Glaz ^{(3)}
 Editor Affiliations

 1. Dipartimento di Matematica, Università degli Studi Roma Tre
 2. Department of Mathematics, Graz University of Technology
 3. Department of Mathematics, University of Connecticut
 Authors

 Byung Gyun Kang ^{(4)}
 Phan Thanh Toan ^{(4)}
 Author Affiliations

 4. Department of Mathematics, Pohang University of Science and Technology, Pohang, 790784, Korea
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