  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.
