Integral domains in which any two \(v\)-coprime elements are comaximal
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}}
scientific article; zbMATH DE number 6377568
scientific article