Chapter

Commutative Algebra

pp 225-238

Date:

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

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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 Non-SFT ring Power series ring Valuation domain

Mathematics Subject Classification (2010):

13A15 13C15 13F05 13F25 13F30.