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 a k ∈ J for each a ∈ I . We prove that for a non-SFT maximal ideal M of an integral domain D , ht ( M 〚 X 〛 / M D 〚 X 〛 ) ≥ 2 ℵ 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 ℵ 1 such that each prime ideal in the chain lies between M D 〚 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 〛 = ht M 〚 X 〛 = ht ( M 〚 X 〛 / M D 〚 X 〛 ) = 2 ℵ 1 for every maximal ideal M of D .