Distributive FCP extensions

Abstract

We are dealing with extensions of commutative rings [Formula: see text] whose chains of the poset [Formula: see text] of their subextensions are finite (i.e. [Formula: see text] has the FCP property) and such that [Formula: see text] is a distributive lattice, that we call distributive FCP extensions. Note that the lattice [Formula: see text] 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 [Formula: see text] distributive [Formula: see text] 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 [Formula: see text] is distributive if and only if [Formula: see text] is distributive, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. A special attention is paid to distributive field extensions.