Pullbacks and universal catenarity

Abstract

This paper deals with the universal catenarity of a pullback construction ring. It seeks necessary and sufficient conditions for such a pullback to have the universal catenarity, improving some known results. Its main result leads to new examples of universally catenarian domains.

A note on maximal non-Noetherian subrings of a domain

“…Because of this, they have played an important role in many research problems in commutative algebra (cf. Anderson et al 1988b;Ayache and Cahen 1992;Jarboui 2005, 2008;Ben Abdallah and Jarboui 2008;Ben Nasr and Jarboui 2002b;Bouvier et al 1988;Cahen 1990;Jarboui and Jerbi 2008).…”

“…A stably strong S-domain is a domain R such that R[n], the ring of polynomials in n indeterminates, is strong S for any n (cf. Jarboui 2005, 2008;Ben Nasr and Jarboui 2002b;Cahen 1990;Jarboui and Jerbi 2008;Kabbaj 1986Kabbaj , 1991Kaplansky 1974;Malik and Mott 1983). Examples of stably strong S-domains are Noetherian and Prüfer domains.…”

“…As this notion does not carry over to localizations and factor rings, R is said to be locally (resp., residually) Jaffard if R P (resp., R/P) is a Jaffard domain for each prime ideal P of R. R is said to be totally Jaffard if R P is residually Jaffard (equivalently R/P is locally Jaffard) for each prime ideal P of R. The class of totally Jaffard domains contains most of the well-known classes of rings involved in Krull dimension theory such as Noetherian domains, Prüfer domains, universally catenarian domains and stably strong S-domains. We assume familiarity with these concepts as in Anderson et al (1988a,b), Ben Nasr and Jarboui (2002b), Bouvier et al (1988), Cahen (1990), Gilmer (1972), Jaffard (1960), Jarboui and Jerbi (2008), Kabbaj (1986Kabbaj ( , 1991, Kaplansky (1974). In Ayache and Jarboui (2002, Theorem 4.5) the authors proved that if R is a maximal non-Noetherian subring of S and S is not a field then R is a totally Jaffard domain.…”