SFT stability via power series extension over Prüfer domains
type: Item
instance of: scholarly article
title: SFT stability via power series extension over Prüfer domains
zbMATH Open document ID: 1166.13021
published in: Manuscripta Mathematica
publication date: 2008-09-25T00:00:00Z
review text: A commutative ring \(D\) with identity is called an SFT ring if for each ideal \(I\) of \(D\), there exist a natural number \(k\) and a finitely generated ideal \(J \subseteq I\) such that \(a^k \in J\) for all \(a \in I\). Let dim\(D\) denote the (Krull) dimension of \(D\), let \(x_1, \dots , x_n\) be indeterminates over \(D\), and let \(Dx_1, \dots , x_n\) be the power series ring over \(D\). It is well known that if dim\((Dx_1) < \infty\), then \(D\) is an SFT ring [\textit{J. T. Arnold}, Trans. Am. Math. Soc. 177, 299--304 (1973; Zbl 0262.13007)]. Also, if \(D\) is a finite dimensional SFT Prüfer domain, then dim\((Dx_1, \dots , x_n) = n (\dim D) +1\), and hence \(Dx_1, \dots , x_n\) is an SFT ring [\textit{J. T. Arnold}, Indiana Univ. Math. J. 31, 897--911 (1982; Zbl 0475.13014)]. However, the power series ring over an SFT ring need not be an SFT ring [\textit{J. Coykendall}, J. Algebra 256, 85--96 (2002; Zbl 1069.13011)]. In this paper, the authors extend Arnold's result to infinite dimensional SFT Prüfer domains. Let \(D\) be an SFT Prüfer domain. They first show that \(Dx_1, \dots , x_n_{D \setminus (0)}\) is an \(n\)-dimensional (noetherian) regular ring, and then they prove that \(Dx_1, \dots , x_n\) is an SFT ring. They also show that if \(P\) is a nonzero prime ideal of \(D\) with ht\(P = m < \infty\), then \[ \text{ht}(Px_1, \dots , x_n) = \begin{cases} mn &\text{if \(P\) is nonmaximal and \(\bar{D}_{\bar{D} \setminus (0)}x_1, \dots , x_n\)}\\ & \neq \bar{D}x_1, \dots , x_n_{\bar{D} \setminus (0)}, \text{ where \(\bar{D} = D/P\)},\\ (m-1)n + 1 &\text{otherwise.} \end{cases} \]
reviewed by: Gyu Whan Chang
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5347175
zbMATH Keywords: (Krull) dimension of a ring
zbMATH Keywords: Prüfer domain
zbMATH Keywords: SFT ring
zbMATH Keywords: power series ring
author: B. G. Kang
author: Mi Hee Park
MaRDI profile type: Publication
full work available at URL: s00229-007-0073-7
OpenAlex ID: W2079037083
cites work: Anti-archimedean rings and power series rings
cites work: Krull Dimension in Power Series Rings
cites work: Power series rings over Prüfer domains
cites work: Algebraic Extensions of Power Series Rings
cites work: The SFT property does not imply finite dimension for power series rings.
cites work: Q5687544
cites work: A Note on the Quotient Field of the Domain DX
cites work: Q5661989
cites work: Q5562323
cites work: Completion of a Prüfer domain
cites work: On Močkor's question
cites work: Q5735274
DOI: 10.1007/S00229-007-0073-7
description: scientific article; zbMATH DE number 5347175