Maximal non-Jaffard subrings of a field

Abstract

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.

When is each proper overring of $R$ an S(Eidenberg)-domain?

“…In [3], the author and M. Ben Nasr considered maximal non-Jaffard subrings of a field L, that is, the domains R where R is a non Jaffard domain and each ring T , R ⊂ T ⊆ L is Jaffard. They characterized these domains in terms of pseudo-valuation domains.…”

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.…”

A note on maximal non-Noetherian subrings of a domain

“…Recall that if P is a ring-theoretic property and R ⊂ S are rings, then R is said to be maximal non-Psubring of S if R does not satisfy P, whereas T satisfies P for each ring T such that R ⊂ T ⊆ S, where the symbol "⊂" denotes proper inclusion. Recent studies of maximal non-P include (Ayache et al 2007; Ayache and Jarboui 2002;Ben Nasr and Jarboui 2000) in which the property P considered by the authors being ACCP, Noetherian or Jaffard, respectively. In Ayache and Jarboui (2002), the authors characterized maximal non-Noetherian subrings in several cases and they showed that if R is local and maximal non-Noetherian subring of S, then R is a pullback of the type R := (S, Q, D), where Q is a prime ideal of S but not necessarily a maximal ideal of S. Recall that a finite-dimensional integral domain R is said to be Jaffard if dim(R[X 1 , .…”