A counterexample for a conjecture about the catenarity of polynomial rings.
type: Item
instance of: scholarly article
title: A counterexample for a conjecture about the catenarity of polynomial rings.
zbMATH Open document ID: 1041.13008
author: M. Ben Nasr
author: Noômen Jarboui
published in: Journal of Algebra
publication date: 2002-07-02T00:00:00Z
review text: It is known that a Noetherian ring \(R\) is universally catenary if \(R[X]\), the polynomial ring in one indeterminate over \(R\), is catenary [\textit{L. J. Ratliff jun.}, Am. J. Math. 92, 99--144 (1970; Zbl 0198.06003)]. The authors show that this is not true in non-Noetherian case. The main result is as follows: Let \(T\) be a Noetherian normal domain with \(\text{height}\,n = 2\) for any maximal ideal \(n\) of \(T\), \(I\) a height one prime ideal of \(T\), \(K\) a field contained in \(T/I\), and \(R\) is the pull-back of \(T \rightarrow T/I \leftarrow K\), that is \(R = \pi^{-1}(K)\) in this case where \(\pi\) is the natural map \(T \to T/I\). Assume that (i) \(\text{trans.deg}\,[T/I:K] \geq 2\) and (ii) \(\text{trans.deg}\,[T/n:K] = 0\) for any maximal ideal \(n\) containing \(I\). Then \(\text{height}\,m = 2\) for any maximal ideal \(m\) of \(R\), \(R[X]\) is catenary and \(R[X,Y]\) is not catenary. For example, let \(R = \mathbb{Z} + p\mathbb{Z}t\) where \(\mathbb{Z}\) is the ring of integers, \(p\) is a prime number and \(t\) is an indeterminate. Then \(R[X]\) is catenary and \(R[X,Y]\) is not catenary.
reviewed by: Yoichi Aoyama
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13E99
zbMATH DE Number: 1749653
zbMATH Keywords: strong S-domain
zbMATH Keywords: altitude formula
zbMATH Keywords: non-Noetherian ring
zbMATH Keywords: non catenary ring
MaRDI profile type: Publication
full work available at URL: jabr.2001.9046
OpenAlex ID: W1966929221
Wikidata QID: Q123025263
cites work: Q4293838
cites work: Universally catenarian integral domains
cites work: Couples d'anneaux partageant un idéal. (Couples of rings sharing an ideal)
cites work: Construction B, I, D et anneaux localement ou residuellement de Jaffard. (B, I, D construction and locally or residually Jaffard rings)
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cites work: Q4405283
cites work: Q5603302
cites work: Strong S-domains
cites work: On Quasi-Unmixed Local Domains, the Altitude Formula, and the Chain Condition for Prime Ideals (II)
DOI: 10.1006/JABR.2001.9046
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description: scientific article; zbMATH DE number 1749653