Uppers to zero in polynomial rings and Pr\"ufer-like domains

Abstract

Let $D$ be an integral domain and $X$ an indeterminate over $D$. It is well known that (a) $D$ is quasi-Pr\"ufer (i.e, its integral closure is a Pr\"ufer domain) if and only if each upper to zero $Q$ in $D[X] $ contains a polynomial $g \in D[X]$ with content $\co_D(g) = D$; (b) an upper to zero $Q$ in $D[X]$ is a maximal $t$-ideal if and only if $Q$ contains a nonzero polynomial $g \in D[X]$ with $\co_D(g)^v = D$. Using these facts, the notions of UM$t$-domain (i.e., an integral domain such that each upper to zero is a maximal $t$-ideal) and quasi-Pr\"ufer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this paper, given a semistar operation $\star$ in the sense of Okabe-Matsuda, we introduce the $\star$-quasi-Pr\"ufer domains. We give several characterizations of these domains and we investigate their relations with the UM$t$-domains and the Pr\"ufer $v$-multiplication domains.