Chains of prime ideals in power series rings

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 subset of I and a positive integer k such that a(k) is an element of J for each a is an element of I. We prove that for a non-SFT maximal ideal M of an integral domain D, ht(M[[X]]/MD[[X]] >= 2(N)1 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 2 & alefsym;1 such that each prime ideal in the chain lies between M[[QX]] /MD[[X]] = x27a2; and MQX2;. 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]) = 2(aleph 1) for every maximal ideal M of D. (C) 2021 Elsevier B.V. All rights reserved.