Strongly comaximizable primes
type: Item
instance of: scholarly article
title: Strongly comaximizable primes
zbMATH Open document ID: 0813.13005
author: Stephen McAdam
published in: Journal of Algebra
publication date: 1995-05-30T00:00:00Z
review text: \textit{W. Heinzer} and \textit{S. Weigand} [Proc. Am. Math. Soc. 107, No. 3, 577-586 (1989; Zbl 0693.13002)] showed that if \(P\) is a nonzero prime ideal in a Noetherian domain \(R\) and if either \(R\) is not Henselian or \(P\) is not maximal, then for any integer \(m \geq 1\), there exists a finitely generated integral extension domain \(T\) of \(R\) in which at least \(m\) primes lie over \(P\). -- \textit{R. Heitmann} and the author [Proc. Am. Math. Soc. 112, No. 3, 661-669 (1991; Zbl 0741.13004)] strengthened this by showing that if \(R\) is not Henselian, then in fact \(T\) can be chosen so that there is a set of \(m\) primes in \(T\) which lie over \(P\) and which are pairwise comaximal. However, in this last result, there may be other primes which lie over \(P\). A prime ideal \(P\) in \(R\) is called strongly comaximizable if for every integer \(m \geq 1\), there is a finitely generated integral extension domain \(T\) of \(R\) such that there are exactly \(m\) primes of \(T\) lying over \(P\) and that those \(m\) primes are pairwise comaximal. In the paper under review the author shows that if \(P\) is not contained in the Jacobson radical of \(R\), then \(P\) is strongly comaximizable. (If \(P\) is contained in the Jacobson radical of \(R\), then \(P\) may or may not be strongly comaximizable.) Moreover, if \(R\) is a nearly Noetherian domain, then the author considers the following problem: Question. Are the concepts of \(H\)-prime and not strongly comaximizable prime equivalent? The author shows that if \(R\) is integrally closed, the answer is yes and if \(R\) is quasi-local, then not strongly comaximizable primes are \(H\)- primes (but he does not know if the converse holds).
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 703028
zbMATH Keywords: strongly comaximizable ideal
zbMATH Keywords: extension domain
zbMATH Keywords: Jacobson radical
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
full work available at URL: jabr.1994.1335
OpenAlex ID: W2074980759
DOI: 10.1006/JABR.1994.1335
Recommended article: Comaximizable Primes
Recommended article: Comaximizing and splitting primes
Recommended article: Independent locally-finite intersections of localizations
Recommended article: Uppers in R[X]
Recommended article: Prime Ideals in Two-Dimensional Polynomial Rings
Recommended article: Q3982316
Recommended article: Integral Closures of Noetherian Integral Domains as Intersections
Recommended article: Localization and right krull links
Recommended article: Strong $n$-generators and the rank of some Noetherian one-dimensional integral domains
Recommended article: Two Notes on Imbedded Prime Divisors
description: scientific article; zbMATH DE number 703028