The Krull dimension of power series rings over almost Dedekind domains

Abstract

Let D be an almost Dedekind domain that is not Dedekind, M be a non-invertible maximal ideal of D, X be an indeterminate over D, and D[X] be the power series ring over D. We first construct an example of eta(1)-sets and we then use this eta(1)-set and M to give a simple proof of dim(D[X]) >= 2(aleph 1). We show that ht(M[X]/MD[X]) >= 2(aleph 1) when D has only countably many non-invertible maximal ideals or when M is countably generated. We finally construct a simple example of almost Dedekind domains that are not Dedekind with given cardinal number of non-invertible maximal ideals. (C) 2015 Elsevier Inc. All rights reserved.