Enlarging localized polynomial rings while preserving their prime ideal structure

摘要

Let n be an integer greater than 1 and let x(1),..., 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 Sbetween the localized polynomial ring k[x(1),..., x(n)](x(1),...,x(n)) and the power series ring k[[x(1),..., x(n)]] such that the prime ideal spectrum of Sis homeomorphic to the prime ideal spectrum of k[x(1),..., x(n)](x(1),...,x(n)) as topological spaces with the Zariski topology (Theorem3.17). Thus Sis a local n-dimensional Noetherian domain and the cardinality of the set of prime ideals of Sis strictly less than the cardinality of S. We also show that every Noetherian ring Awith infinitely many prime ideals has a Noetherian subring Bsuch that the prime ideal spectrum of Bis homeomorphic to the prime ideal spectrum of Aand the cardinality of the set of prime ideals of Bequals the cardinality of B. (C) 2021 Elsevier Inc. All rights reserved.