Around Prufer extensions of rings

Abstract

The paper intends to apply the properties of Pr\"ufer extensions, investigated in the Knebusch-Zhang book, to ring extensions $R\subseteq S$. The integral closure $\overline R$ of $R$ in $S$ is shown to be the intersection of all $T\in [R,S]$, such that $T\subseteq S$ is Pr\"ufer. We are then able to establish an avoidance lemma for integrally closed subextensions. Rings of sections of the affine scheme defined by $R$ provide results on $S$-regular ideals. Some results on pullbacks characterizations of Pr\"ufer extensions are given. We introduce locally strong divisors, examining the properties of strong divisors of a local ring and their links with Pr\"ufer extensions. The locally strong divisors allow us to give characterizations of QR-extensions. We apply our results to Nagata extensions of rings. We also look at the Pr\"ufer hull of a Nagata extension. We define quasi-Pr\"uferian rings that may differ from quasi-Pr\"ufer integral domains. We then derive some results on minimal and FCP extensions. Finally, we study the set of all primitive elements in an extension.