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⊆I and a positive integer k such that ak∈J for all a∈I. For a cardinal number α and a ring D, we say that dim⁡D≥α if D has a chain of prime ideals with length ≥α. Arnold showed that if D is a non-SFT ring, then dim⁡D〚X〛≥ℵ0. 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〛≥2ℵ1 for every D∈C and that under the continuum hypothesis 2ℵ1 is the greatest lower bound of dim⁡D〚X〛 for D∈C. On the other hand, there exists a (finite-dimensional) SFT domain D such that dim⁡D〚X〛≥2ℵ1.