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

چکیده مقاله

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.