Locally factorial integral domains
An integral domain R is locally factorial if \(R_ f=R[1/f]\) is factorial for each nonzero nonunit f of R. A non quasi-local locally factorial domain is a Krull domain. Claborn gave examples of locally factorial Dedekind domains with cyclic divisor class groups (and with the further property that every proper overring is a PID). We show that if R is a locally factorial Krull domain, then \(Cl(R)={\mathbb{Z}}^ n\oplus {\mathbb{Z}}/m{\mathbb{Z}}\) where \(n\geq 0\), \(m\geq 1\). Conversely, for each such n and m we construct a locally factorial Dedekind domain R with \(Cl(R)={\mathbb{Z}}^ n\oplus {\mathbb{Z}}/m{\mathbb{Z}}.\) We also give an example of a locally factorial Dedekind domain that has an overring which is not locally factorial. Among the many other results, we show that if R is a Krull domain with finitely generated nonzero divisor class group, then some localization of R is locally factorial but not factorial. Quasi- local locally factorial domains that are not Krull domains are also investigated.
finitely generated nonzero divisor class group
scientific article; zbMATH DE number 3867471
scientific article