Universally Catenarian Integral Domains, Strong S-Domains and Semistar Operations

Abstract

Let D be an integral domain and ★ a semistar operation stable and of finite type on it. In this article, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over D. We introduce and investigate the notions of ★-universally catenarian and ★-stably strong S-domains and prove that, every ★-locally finite dimensional Prüfer ★-multiplication domain is ★-universally catenarian, and this implies ★-stably strong S-domain. We also give new characterizations of ★-quasi-Prüfer domains introduced recently by Chang and Fontana, in terms of these notions.

Key Words:

• Catenary
• Jaffard domain
• Krull dimension
• Semistar operation
• Star operation
• Strong S-domain
• Universally catenarian

2000 Mathematics Subject Classification:

• Primary 13G05, 13B24, 13A15, 13C15

ACKNOWLEDGMENT

I would like to thank Professor Marco Fontana for his comments on this article. I also thank the referee for several helpful remarks concerning the final form of the article.

Notes

Communicated by I. Swanson.