# Boolean FIP ring extensions

## TL;DR

This paper characterizes Boolean FIP ring extensions, where the subextensions form a finite Boolean lattice, and shows that all subextensions are simple, revealing structural properties of such extensions.

## Contribution

It provides a characterization of Boolean FIP ring extensions and demonstrates that all subextensions are simple, advancing understanding of their lattice structure.

## Key findings

- Subextensions of Boolean FIP extensions are simple.
- Characterizations involve factorial properties of the subextension poset.
- Each Boolean FIP extension has a finite Boolean lattice of subextensions.

## Abstract

We characterize extensions of commutative rings $R \subseteq S$ whose sets of subextensions $[R,S]$ are finite ({\it i.e.} $R\subseteq S$ has the FIP property) and are Boolean lattices, that we call Boolean FIP extensions. Some characterizations involve ``factorial" properties of the poset $[R,S]$. A non trivial result is that each subextension of a Boolean FIP extension is simple (i.e. $R \subseteq S$ is a simple pair).