Residually algebraic pairs of rings

Abstract

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

Some finiteness conditions on the set of overrings of an integral domain

“…We use the term FO-domain or FC-domain to refer to an integral domain satisfying the respective condition (FO) or (FC). In extending results of [2] in [13] and [14], Jaballah asked [14, Quest. 1] for a characterization of FO-domains.…”

“…Suppose D is an integral domain with quotient field K. In their study of residually algebraic pairs of integral domains in [2], Ayache and Jaballah encountered the following two conditions on the set of overrings 1 of D; we label these as (FO) and (FC):…”

Ring extensions with some finiteness conditions on the set of intermediate rings

“…Recall that a ring extension R ⊆ T is called a residually algebraic extension, [5], if for each prime ideal Q of T , T /Q is algebraic over R/(Q ∩ R). We say that (R, S) is a residually algebraic pair, Definition 2.1 of [1], if for each T ∈ [R, S], R ⊆ T is a residually algebraic extension. We say that (R, S) is a normal pair, [4], if for each T ∈ [R, S], T is integrally closed in S. The extension R ⊆ S is called a primitive extension (or P-extension, see [8]) if each element u of S is a root of a polynomial f (X) ∈ R[X] with unit content; i.e., the coefficients of f (X) generate the unit ideal of R.…”

“…Residually algebraic pairs (R, S) with R integrally closed in S are necessarily normal pairs by Theorem 2.5 (vi) of [1], therefore they enjoy several nice properties. The following properties of normal pairs are going to be used in this paper.…”

“…(3) (Lemma 3.1 of [1]) Max(T ) is the subset of maximal elements in the set {Q i T : i ∈ I}. (4) Each prime ideal of T is an extension P T of a prime ideal P of R. Furthermore, for each prime Q of R such that QT = T , QT is a prime ideal of T satisfying QT ∩ R = Q and T QT = R QT ∩R = R Q .…”