Idempotence and divisorialty in Pr\"ufer-like domains

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 $D$.