Catenarian FCP ring extensions

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–Hölder 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 types of extensions are catenarian. We reduce the characterization of catenarian extensions to the case of field extensions, an unsolved question at this time.