A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dimv 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.

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

For a ring extension R ⊆ S, (R, S) is called a universally catenarian pair if every domain T , R ⊆ T ⊆ 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] ≤ 1. For dimR ≥ 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. (1991):13B02, 13C15, 13A17, 13A18, 13B25, 13E05 Mathematics Subject Classification IntroductionThroughout this paper, R → S denotes an extension of commutative rings, qf (R) the quotient field of R; tr.deg[S : R] the transcendence degree of qf (S) over qf (R). We use the notations in [10] and [11]. In particular, ⊂ denotes proper containment and ⊆ denotes containment with possible equality.Let P be a ring theoretic property and R ⊆ S an extension of commutative rings. We say that (R, S) is a P-pair if each intermediate ring between R and S satisfies the property P. A crucial problem concerning pairs of rings is to give necessary and sufficient conditions in order to provide a P-pair.Given a finite dimensional ring R, we say that R is a Jaffard ring if dimR = dim v R [2]. The previous property is not local and thus we say

This paper solves a long-standing open question: it is known that, if R is a Noetherian ring such that R X is catenarian, then so is R X Y , and, hence, R is universally catenarian; yet the non-Noetherian case remains unsolved. We do provide here an answer with a two-dimensional coequidimensional counterexample.

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