Ascending the divided and going-down properties by absolute flatness
An \textit{absolutely flat ring morphism} (a notion introduced by J. P. Olivier) is a flat ring morphism \(R\to S\) such that the codiagonal morphism \(S\otimes_R S \to S\) is flat. Examples are (strict) henselizations and étale morphisms. In the paper under review the author is primarily interested in showing that the ``divided ring'' and ``going-down ring'' properties ascend along absolutely flat ring morphisms. The author is motivated by some papers by M. G. Cinquegrani where mainly the ascent by henselization of the ``divided domain'' property is considered, and, as a corollary, by integral extensions. The author extends some of Cinquegrani's results to rings which may not be integral domains and absolutely flat morphisms. He also gives ascent results for the ``going-down ring'' property by using some recent results of \textit{A. Badawi} and \textit{D. E. Dobbs} [Commun. Algebra 29, 2805--2825 (2001; Zbl 1104.13301)]. As a by-product he obtains new results about the ascent of the ``going-down ring'' property by integral extensions. All the results are established under either a unibranchedness hypothesis or at least an irreducibility condition. The consideration of unibranchedness is necessary because, for instance, if \(R^h\) is divided, then \(R\) is unibranched and \(R\to R^h\) is absolutely flat.
scientific article; zbMATH DE number 6116569
scientific article