All rings considered in this paper are supposed to be integral domains. The ÿeld of fractions of a ring R is denoted k(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, tr.deg[S : R] the transcendence degree of S over R, and by R * the integral closure of R in S. If the set [R; S] is ÿnite, we will denote by |[R; S]| its cardinality. If P and Q are two primes of R such that P ⊆ Q; then [P; Q] denote the set of all primes˝of R such that P ⊆˝⊆ Q:The main purpose of this paper is to study residually algebraic extensions and pairs. The ring extension R ⊂ S is said to be residually algebraic, if for each prime Q of S and P = Q ∩ R; the extension R=P ⊂ S=Q is algebraic. The pair of rings (R; S) is said to be residually algebraic if for each ring T in [R; S]; the extension R ⊂ T is residually algebraic. This kind of extensions were ÿrst considered by D.E. Dobbs and M. Fontana to characterize homomorphisms of universally incomparable rings [6]. They were also recently considered by A. Ayache and P.J. Cahen [3] for the study of rings R such that the extension R ⊂ T satisÿes the dimension inequality for each overring T of R; i.e. T ∈ [R; k(R)]:In the ÿrst and second sections, we settle the elementary properties of residually algebraic extensions and pairs. We also compare them with extensions that are going up, incomparable, lying over, integral, going down, satisfying the dimension formula or the dimension inequality, Propositions 1.5 and 1.6, Corollary 1.7, Theorem 2.3, Corollary 2.11, and Theorem 2.12. We give also several characterizations of residually algebraic pairs. We show that such pairs satisfy the (u; u −1 )-lemma, which enable us to give a surprising generalization of this lemma, Theorem 2.5. A global version of this characterization is also settled, Theorem 2.10. In the third section we impose some ÿniteness conditions on the ring R, such as to be semilocal and of ÿnite

No abstract

No abstract

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.

No abstract