Universally catenarian and going-down pairs of rings

Abstract

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

A note on maximal non-Noetherian subrings of a domain

“…We recall that a pair of rings (A, B) where A ⊆ B is an extension of integral domains is said to be a universally catenarian pair if each ring T such that A ⊆ T ⊆ B is universally catenarian (cf. Ayache et al 2001; Ben Abdallah and Jarboui 2008). Next we prove that if R is a maximal non-Noetherian subring of S, then R is universally catenarian iff S is universally catenarian iff (R, S) is a universally catenarian pair.…”

Pairs of Domains Where All Intermediate Domains are Treed

“…The titular result of this section is the following theorem: Proof Let R and S be, respectively, the integral closures of R and S. According to [4,Corollary 5.28], R is a Prüfer domain. If R 1 is the integral closure of R in q f (S), then every element of R 1 is integral over R, so over S. We have then the inclusions R ⊂ R ⊂ R 1 ⊂ S .…”

Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

“…Examples of such systems are the family of all S-overrings, that of all proper S-overrings, etc. (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). For a ring-theoretic property B and a ring extension R ⊂ S, let C denote the family of S-overrings T of R such that T does not satisfy B.…”