How to Construct Huge Chains of Prime Ideals in Power Series Rings

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.