Some remarks on Pr\"{u}fer $\star$--multiplication domains and class groups

Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$. Also, let $\boldsymbol{$\mathcal{T}$}:={T_{\lambda}\mid \lambda \in \Lambda }$beadefiningfamilyofquotientringsof$D$ and suppose that $\ast$ is a finite type star operation on $D$ induced by $\mathcal{T}$. We show that $D$ is a P$\ast$MD (resp., P$v$MD) if and only if $({\text{co}}_D(fg){)}^{\ast }=({\text{co}}_D(f){\text{co}}_D(g){)}^{\ast }$ (resp., $({\text{co}}_D(fg){)}^w=({\text{co}}_D(f){\text{co}}_D(g){)}^w$) for all $0\mathrm{\ne }f,g\in K[X]$. A more general version of this result is given in the semistar operation setting. We give a method for recognizing P$v$MD’s which are not P$\ast$MD’s for a certain finite type star operation $\ast$. We study domains $D$ for which the $\ast$–class group ${\text{Cl}}^{\ast }(D)$ equals the $t$–class group ${\text{Cl}}^t(D)$ for any finite type star operation $\ast$, and we indicate examples of P$v$MD’s $D$ such that ${\text{Cl}}^{\ast }(D)⊊{\text{Cl}}^t(D)$. We also compute ${\text{Cl}}^v(D)$ for certain valuation domains $D$.