Around Prufer extensions of rings

TL;DR

This paper explores properties of Pr"ufer extensions in ring theory, establishing new characterizations, avoidance lemmas, and applying these to Nagata extensions and primitive elements, advancing the understanding of ring extension structures.

Contribution

It introduces new characterizations of Pr"ufer extensions, including avoidance lemmas and links with strong divisors, and applies these to Nagata extensions and primitive elements.

Findings

Integral closure as intersection of Pr"ufer subextensions

Characterization of QR-extensions via strong divisors

Results on minimal and FCP extensions

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.