Chains of prime ideals in power series rings

Abstract

An ideal I of a commutative ring D with identity is called an SFT ideal if there exist a finitely generated ideal J with J⊆I and a positive integer k such that ak∈J for each a∈I. We prove that for a non-SFT maximal ideal M of an integral domain D, ht(M〚X〛/MD〚X〛)≥2ℵ1 if either (1) D is a 1-dimensional quasi-local domain (in particular D is a 1-dimensional nondiscrete valuation domain) or (2) M is the radical of a countably generated ideal. In other words, if one of the conditions (1) and (2) is satisfied, then there is a chain of prime ideals in D〚X〛 with length at least 2ℵ1 such that each prime ideal in the chain lies between MD〚X〛 and M〚X〛. As an application, assuming the continuum hypothesis we show that if D is either the ring of algebraic integers or the ring of integer-valued polynomials on Z, then dim⁡D〚X〛=htM〚X〛=ht(M〚X〛/MD〚X〛)=2ℵ1 for every maximal ideal M of D. © 2021 Elsevier B.V.