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

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.