Authors’ abstract: Given a power series ring \(R^{*}\) over a Noetherian integral domain \(R\) and an intermediate field \(L\) between \(R\) and the total quotient ring of \(R^{*}\), the integral domain \(A = L \cap R^{*}\) often (but not always) inherits nice properties from \(R^{*}\) such as the Noetherian property. For certain fields \(L\) it is possible to approximate \(A\) using a localization \(B\) of a particular nested union of polynomial rings over \(R\) associated to \(A\); if \(B\) is Noetherian, then \(B = A\). If \(B\) is not Noetherian, we can sometimes identify the prime ideals of \(B\) that are not finitely generated. We have obtained in this way, for each positive integer \(m\), a three-dimensional local unique factorization domain \(B\) such that the maximal ideal of \(B\) is two-generated, \(B\) has precisely \(m\) prime ideals of height 2, each prime ideal of \(B\) of height 2 is not finitely generated and all the other prime ideals of \(B\) are finitely generated. We examine the structure of the map \(\mathrm{Spec} A\rightarrow \mathrm{Spec}B\) for this example. We also present a generalization of this example to dimension four. This four-dimensional, non-Noetherian local unique factorization domain has exactly one prime ideal \(Q\) of height three, and \(Q\) is not finitely generated.