Universally catenarian and going-down pairs of rings

Abstract.

For a ring extension $R\subseteq S, (R, S)$ is called a universally catenarian pair if every domain $T, R\subseteq T\subseteq S$, is universally catenarian. When R is a field it is shown that the only universally catenarian pairs are those satisfying $tr.deg[S:R]\leq 1$. For $dimR\geq 1$ several satisfactory results are given. The second purpose of this paper is to study going-down pairs (Definition 5.1). We characterize these pairs of rings and we establish a relationship between universally catenarian, going-down and residually algebraic pairs.