Catenarian FCP ring extensions

TL;DR

This paper studies the catenarian property of ring extensions, which relates to the Jordan-H"older property, and reduces the problem to the case of field extensions, providing new insights into their structure.

Contribution

It extends the study of catenarian extensions from finite field cases to arbitrary ring extensions, linking the property to field extension cases.

Findings

Many types of extensions are catenarian

Reduction of the problem to field extensions

Provides new criteria for catenarian property

Abstract

If $R\subseteq S$ is a ring extension of commutative unital rings, the poset $[R,S]$ of $R$-subalgebras of $S$ is called catenarian if it verifies the Jordan-H\"older property. This property has already been studied by Dobbs and Shapiro for finite extensions of fields. We investigate this property for arbitrary ring extensions, showing that many type of extensions are catenarian. We reduce the characterization of catenarian extensions to the case of field extensions, an unsolved question at that time.