Krull dimension of power series rings over non-SFT domains

Abstract

A ring D is called an SFT ring if for each ideal I of D, there exist a finitely generated ideal J of D with J subset of I and a positive integer k such that a(k) is an element of J for all a is an element of I. For a cardinal number alpha and a ring D, we say that dim D > alpha if D has a chain of prime ideals with length >= alpha. Arnold showed that if D is a non-SFT ring, then dim D[X] >= N0. Let C be the class of non-SFT domains. The class C includes the class of finite -dimensional nondiscrete valuation domains, the class of non-Noetherian almost Dedekind domains, the class of completely integrally closed domains that are not Krull domains, the class of integral domains with non-Noetherian prime spectrum, and the class of integral domains with a nonzero proper idempotent ideal. The ring of algebraic integers, the ring of integer -valued polynomials on Z, and the ring of entire functions are also members of the class C. In this paper we prove that dim D[X] >= 2(N1) for every D is an element of C and that under the continuum hypothesis 2(N1) is the greatest lower bound of dim D[X] for D is an element of C. On the ther hand, there exists a (finite-dimensional) SFT domain D such that dim D[X] >= 2(N1). (C) 2017 Elsevier Inc. All rights reserved.