Distributive FCP extensions

TL;DR

This paper studies distributive FCP extensions of commutative rings, characterizing when such extensions are distributive, especially focusing on the relation to their integral closures and specific cases like field extensions.

Contribution

It provides a characterization of distributive FCP extensions and explores their properties, including the relation to integral closures and special cases like field extensions.

Findings

Distributive FCP extensions are characterized by their integral closure.

The lattice of subextensions in a distributive FCP extension is finite.

Special results are obtained for distributive field extensions.

Abstract

We are dealing with extensions of commutative rings $R\subseteq S$ whose chains of the poset $[R,S]$ of their subextensions are finite ({\em i.e.} $R\subseteq S$ has the FCP property) and such that $[R,S]$ is a distributive lattice, that we call distributive FCP extensions. Note that the lattice $[R,S]$ of a distributive FCP extension is finite. This paper is the continuation of our earlier papers where we studied catenarian and Boolean extensions. Actually, for an FCP extension, the following implications hold: Boolean $\Rightarrow$ distributive $\Rightarrow$ catenarian. A comprehensive characterization of distributive FCP extensions actually remains a challenge, essentially because the same problem for field extensions is not completely solved. Nevertheless, we are able to exhibit a lot of positive results for some classes of extensions. A main result is that an FCP extension $R\subseteq S$ is distributive if and only if $R\subseteq\overline R$ is distributive, where $\overline R$ is the integral closure of $R$ in $S$. A special attention is paid to distributive field extensions.