Divisibility properties related to star-operations on integral domains
The authors introduce and study the notion of \(GCD\)-Bézout domains. A domain \(R\) is called a \(GCD\)-Bézout domain if the Bézout identity holds for any set of nonzero elements of \(R\) whose \(\mathrm{gcd}\) exists. Among others, they characterize \(GCD\)-Bézout domains as \(DW\)-domains (i. e., the \(w\)-operation is trivial, or \(w=d\)) having the \(PSP\)-property (that is, each primitive ideal is superprimitive). They also introduced a new (semi)-star operation called the \(\tilde{p}\)-operation and used it to give another characterization of \(GCD\)-Bézout domains. It turns out that a domain is a \(GCD\)-Béezout domain if and only if \(\tilde{p}=d\) as star operation.
scientific article; zbMATH DE number 6098603