Constructing chains of primes in power series rings

Abstract

For an integral domain D of dimension n, the dimension of the polynomial ring D [ x ] is known to be bounded by n + 1 and 2 n + 1 . While n + 1 is a lower bound for the dimension of the power series ring D [ [ x ] ] , it often happens that D [ [ x ] ] has infinite chains of primes. For example, such chains exist if D is either an almost Dedekind domain that is not Dedekind or a rank one nondiscrete valuation domain. One concern here is developing schemes by which such chains can be constructed in D [ [ x ] ] when D is an almost Dedekind domain. A consequence of these constructions is that there are chains of primes similar to the set of ω 1 transfinite sequences of 0ʼs and 1ʼs ordered lexicographically.