Krull dimension of a power series ring over a nondiscrete valuation domain is uncountable

Abstract

Let V be a rank-one nondiscrete valuation domain with maximal ideal M. We prove that the Krull-dimension of V[X](v\(0)) is uncountable, and hence the Krull-dimension of V[X] is uncountable. This corresponds to the well-known fact that the Krull-dimension of the ring of entire functions is uncountable. In fact we construct an uncountable chain of prime ideals inside M[X] such that all the members contract to (0) in V. Our method provides a new proof that the Krull-dimension of the ring of entire functions is uncountable. It is also shown that V[X](v\(0)) is not even a Prufer domain, while the ring of entire functions is a Bezout domain. These are answers to Eakin and Sathaye's questions. Applying the above results, we show that the Krull-dimension of V[X] is uncountable if V is a nondiscrete valuation domain. (C) 2012 Elsevier Inc. All rights reserved.