Krull dimension of power series rings

Abstract

An ideal I of a commutative ring R with identity is called an SFT (strong finite type) ideal if there exist a finitely generated ideal J of R 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. A ring R is called an SFT ring if every ideal of R is an SFT ideal. Arnold showed that if R is a non-SFT ring, then the Krull dimension of the power series ring R[X] is infinite. Let C be the class of non-SFT rings. In this paper, we show that dim R[X] >= 2(N1) for every R is an element of C. In other words, if R is a non-SFT ring, then there exists a chain of prime ideals in R[X] with length at least 2(N1). We also prove that under the continuum hypothesis 2(N1) is the greatest lower bound of dim R[X] for R is an element of C. If M is a non-SFT maximal ideal of a ring R such that M is the radical of a countably generated ideal, then we construct a chain {P-alpha} of prime ideals in R[X] with length at least 2(N1) lying between MR[X] and M[X], i.e., MR[X] subset of P-alpha subset of M[X] for each alpha. The same result holds when M is the non-SFT maximal ideal of a zero-dimensional quasi-local ring or a one-dimensional quasi-local domain R. (C) 2020 Elsevier Inc. All rights reserved.