Enlarging localized polynomial rings while preserving their prime ideal structure

Abstract

Let n be an integer greater than 1 and let x1,…,xn 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[x1,…,xn](x1,…,xn) and the power series ring k[[x1,…,xn]] such that the prime ideal spectrum of S is homeomorphic to the prime ideal spectrum of k[x1,…,xn](x1,…,xn) 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.