  Let $D$ be a Pr\"ufer $\star$-multiplication domain, where $\star$ is a
semistar operation on $D$. We show that certain ideal-theoretic properties
related to idempotence and divisoriality hold in Pr\"ufer domains, and we use
the associated semistar Nagata ring of $D$ to show that the natural
counterparts of these properties also hold in $D$.
