The main purpose of this paper is to study maximal non-Noetherian subrings R of a domain S. We give characterizations of such domains in several cases. If the ring R is semi-local, R S is a residually algebraic pair and R is a maximal nonNoetherian subring of S, we give sharp upper bounds for the number of rings and the length of chains of rings in R S , the set of intermediary rings. Key Words: Jaffard domain; Krull dimension; valuation domain; Noetherian domain; finite-type module; pullbacks. INTRODUCTIONAll rings considered in this paper are supposed to be integral domains. The quotient field of a ring R is denoted by qf R , the Krull dimension dim R, and the integral closure R . For a ring extension R ⊂ S, we denote by R S the set of all rings T such that R ⊆ T ⊆ S, by tr deg S R the transcendence degree of qf S over qf R , and by R * the integral closure of R in S. If the set R S is finite, we will denote by R S its cardinality. If P and Q are two primes of R such that P ⊆ Q, then P Q denotes the set of all primes Q of R such that P ⊆ Q ⊆ Q. Let R be a domain; we denote by R n the ring of polynomials in n indeterminates with coefficient in R, (for n = 1, R 1 = R X is the ring of polynomials in one indeterminate). We recall that a ring R of finite (Krull) dimension n is a Jaffard ring if its valuative dimension (the limit of the sequence dim R X 1 X n − n, All rights reserved.
Let [Formula: see text] be a (unital) extension of (commutative) rings. We say that [Formula: see text] is a maximal non-quasi-local (respectively, non-PVD) subring of [Formula: see text] if [Formula: see text] is not quasi-local (respectively, PVD) and each subring of [Formula: see text] properly containing [Formula: see text] is quasi-local (respectively, PVD). The aim of this paper is to study this kind of ring extensions and to investigate the structure of the intermediate rings between [Formula: see text] and [Formula: see text].
Communicated by E.M. Friedlander MSC: Primary: 13B02 secondary: 13C15 13A17 13A18 13B25 13E05 a b s t r a c t We establish in this paper a result that gives the number of intermediary rings between R and S where (R, S) is a normal pair of rings. This result answers in particular a question which was left open in [A. Jaballah, Finiteness of the set of intermediary rings in a normal pair, Saintama Math. J. 17 (1999) 59-61]. Further applications are also given.
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.
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