Universally catenarian integral domains, strong S-domains and semistar operations

TL;DR

This paper explores the dimension theory of polynomial rings over integral domains with semistar operations, introducing new classes of domains and characterizing their properties in the context of semistar operations.

Contribution

It introduces the concepts of rb1-unib1versally catenarian and rb1-stably strong S-domains under semistar operations, providing new characterizations of rb1-quasi-Prb1fer domains.

Findings

rb1-locally finite dimensional Prb1fer rb1-multiplication domains are rb1-universally catenarian.

rb1-universally catenarian domains imply rb1-stably strong S-domains.

New characterizations of rb1-quasi-Prb1fer domains are provided.

Abstract

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