Author’s abstract: Let \(n\) be an integer greater than \(1\) and let \(x_1,\dots, x_n\) be indeterminates over a countable field \(k\). In this paper, we employ techniques of Heitmann and Nagata to show there exists an uncountable regular local ring \(S\) between the localized polynomial ring \(k[x_1,\dots, x_n]_{(x_1,\dots,x_n)}\) and the power series ring \(k[[x_1,\dots, x_n]]\) such that the prime ideal spectrum of \(S\) is homeomorphic to the prime ideal spectrum of \(k[x_1,\dots, x_n]_{(x_1,\dots,x_n)}\) as topological spaces with the Zariski topology (Theorem 3.17). Thus \(S\) is a local \(n\)-dimensional Noetherian domain and the cardinality of the set of prime ideals of \(S\) is strictly less than the cardinality of \(S\). We also show that every Noetherian ring \(A\) with infinitely many prime ideals has a Noetherian subring \(B\) such that the prime ideal spectrum of \(B\) is homeomorphic to the prime ideal spectrum of \(A\) and the cardinality of the set of prime ideals of \(B\) equals the cardinality of \(B\).