The Krull dimension of power series rings over almost Dedekind domains

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 η1-sets and we then use this η1-set and M to give a simple proof of dim⁡(D〚X〛)≥2ℵ1. We show that ht(M〚X〛/MD〚X〛)≥2ℵ1 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.