Maximal non-Jaffard subrings of a field

A domain $R$ is called a maximal non-Jaffard subring of a field $L$ if $R\subset L$, $R$ is not a Jaffard domain and each domain $T$ such that $R\subset T\subseteq L$ is Jaffard. We show that maximal non-Jaffard subrings $R$ of a field $L$ are the integrally closed pseudo-valuation domains satisfying $\dim_v R = \dim R + 1$. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when $R$ is integrally closed. Moreover, these domains are characterized in terms of the altitude formula in case $R$ is not integrally closed. An example of a maximal non-universally catenarian subring of its quotient field which is not integrally closed is given (Example 4.2). Other results and applications are also given.