Krull dimension of mixed extensions
type: Item
instance of: scholarly article
title: Krull dimension of mixed extensions
zbMATH Open document ID: 1180.13015
published in: Journal of Pure and Applied Algebra
publication date: 2009-06-18T00:00:00Z
review text: Let \(R\) be a commutative ring. Then it is well known that \(\dim R+1\leq \dim RX\leq 2\dim R+1\). What happens if we replace \(R[X]\) by \(RX\)? Certainly \(\dim RX\geq \dim R+1\) and we have equality if \(R\) is Noetherian. Recall that \(R\) is an \textit{SFT ring} if for each ideal \(I\) of \(R\), there exists a finitely generated ideal \(B\subseteq I\) and a positive integer \(k\) depending on \(I\) such that \(a^{k}\in B\) for each \(a\in I\). \textit{J. T. Arnold} [Trans. Am. Math. Soc. 177, 299--304 (1973; Zbl 0262.13007)] has shown that if \(R\) is non SFT, then \(\dim RX=\infty \) and \(\dim D[[X_{1},\dots,X_{n}]]=n\dim D+1\) if \(D\) is an SFT Prüfer domain [\textit{J. T. Arnold}, Indiana Univ. Math. J. 31, 897--911 (1982; Zbl 0475.13014)]. \textit{J. Coykendall} [J. Algebra 256, No. 1, 85--96 (2002; Zbl 1069.13011)] gave an example of a finite dimension SFT ring with \(\dim RX=\infty \) and raised the question (also raised by Gilmer) of whether \(\dim RX<\infty \) implies that \(\dim RX\leq 2\dim R+1\). This paper shows that the answer is no. This paper considers ``mixed extensions'' \(R[X_{1}]]\cdots [ X_{n}]]\) where \([X]]\) means that \([X]]\) could either be \([X]\) or \(X\). Generalizing Arnold's result they show that for \(D\) a finite dimensional SFT Prüfer domain (e.g., a finite dimensional discrete valuation domain) \(\dim D[X_{1}]]\cdots [ X_{n}]]=n\dim D+1\) provided at least one of the extensions is a power series extension (of course, for \(D\) Prüfer we have \( \dim D[X_{1},\dots,X_{n}]=\dim D+n\)). In particular, if \(R=D[X_{1},\dots,X_{n-1}]\) and \(\dim D=d>2(n-1)/(n-2)\) (where \(n>2\)), then \(\dim RX=dn+1>2\dim R+1.\)
reviewed by: D. D. Anderson
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 5566280
zbMATH Keywords: Krull dimension
zbMATH Keywords: power series ring
zbMATH Keywords: Prüfer domain
author: B. G. Kang
author: Mi Hee Park
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.02.010
OpenAlex ID: W2031880026
cites work: Krull Dimension in Power Series Rings
cites work: Power series rings over Prüfer domains
cites work: Power series rings over discrete valuation rings
cites work: Q3931523
cites work: Q3933092
cites work: The SFT property does not imply finite dimension for power series rings.
cites work: Q4466611
cites work: SFT stability via power series extension over Prüfer domains
cites work: Q5661989
cites work: Q4044675
cites work: Q5735274
cites work: Q3739243
cites work: Q5603302
cites work: Anti-archimedean rings and power series rings
cites work: Algebraic Extensions of Power Series Rings
DOI: 10.1016/J.JPAA.2009.02.010
Recommended article: A localization of a power series ring over a valuation domain
Recommended article: SFT stability via power series extension over Prüfer domains
Recommended article: Dimension de Krull des anneaux de séries formelles sur un produit fibré. (Krull dimension of formal power series rings on a fibre product)
Recommended article: The SFT property does not imply finite dimension for power series rings.
Recommended article: The Krull dimension of power series rings over non-SFT rings
Recommended article: CONSTRUCTING CHAINS OF PRIMES IN POWER SERIES RINGS, II
Recommended article: Constructing chains of primes in power series rings
Recommended article: Krull dimension of power series rings over non-SFT domains
Recommended article: An Elementary Proof for the Krull Dimension of a Polynomial Ring
Recommended article: Q3436761
description: scientific article; zbMATH DE number 5566280