Splitting ring extensions

TL;DR

This paper investigates the structure of ring extensions, focusing on FCP extensions, introducing the concept of splitting elements, and providing combinatorial results including explicit Prufer hull computations.

Contribution

It introduces and studies splitting elements in ring extensions, especially for FCP extensions, and explores their role in constructing splitters and computing Prufer hulls.

Findings

Split extensions cannot be pinched unless trivial.

Explicit formulas for Prufer hulls of FCP extensions.

Conditions for the existence of splitters in integral extensions.

Abstract

The paper deals with ring extensions $R\subseteq S$ and their lattices $[R,S]$ of subextensions and is mainly devoted to FCP extensions (extensions whose lattices are Artinian and Noetherian). The object of the paper is the introduction and the study of elements of the lattices that split in some sense ring extensions. The reason why is that this splitting was used in earlier paper without their common nature being recognized. There are some favorable cases allowing to build splitters, mainly when we are dealing with $\mathcal B$-extensions, for example integral extensions. Integral closures and Prufer hulls of extensions play a dual role. The paper gives many combinatorics results with the explicit computation of the Prufer hull of an FCP extension. We show that a split extension cannot be pinched, except trivially.