Integral domains in which any two \(v\)-coprime elements are comaximal
type: Item
instance of: scholarly article
title: Integral domains in which any two \(v\)-coprime elements are comaximal
zbMATH Open document ID: 1309.13005
author: Muhammad Zafrullah
author: Evan Houston
published in: Journal of Algebra
publication date: 2014-12-05T00:00:00Z
review text: Let \(D\) be an integral domain. Two nonzero elements \(a, b \in D\) are called \(v\)-coprime if \((a, b)_v = D\) (i.e., \(aD \cap bD = abD\) or equivalently \((a, b)^{-1} = D\)) and comaximal if \((a, b) = D\), where \(v\) is the well-known \(v\)-operation. This paper deals mainly with DF-domains, that is, domains in which any two \(v\)-coprime elements are comaximal. Here are some of the results proved in the paper: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] A domain \(D\) is a DF-domain if and only if the \(d\)- and \(F\)-operations on \(D\) are identical, where \(d\) is the trivial star operation and \(F\) is the star operation defined by \textit{H. E. Adams} [Pac. J. Math. 66, 1--8 (1976; Zbl 0359.13003)]. \item[{\(\bullet\)}] For a domain \(D\), if \(D_M\) is a DF-domain for each maximal ideal \(M\) of \(D\), then \(D\) is a DF-domain. \item[{\(\bullet\)}] Every ideal of a DF-domain has the classical grade at most one. \item[{\(\bullet\)}] For a domain \(D\), the polynomial ring \(D[X]\) is a DF-domain if and only if \(D\) is a field. \end{itemize}}
reviewed by: Fanggui Wang
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 6377568
zbMATH Keywords: star operation
zbMATH Keywords: w-operation
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.10.006
OpenAlex ID: W2317504137
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DOI: 10.1016/J.JALGEBRA.2014.10.006
description: scientific article; zbMATH DE number 6377568