Some remarks on Prüfer ⋆-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 T:={Tλ|λ∈Λ} be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by T. We show that D is a P*MD (respectively, PvMD) if and only if (cD*(fg))=(cD(f)cD*(g)) (respectively, (cDw(fg))=(cD(f)cDw(g))) for all 0≠f,g∈K[X]. A more general version of this result is given in the semistar operation setting. We give a method for recognizing PvMD's which are not P*MD's for a certain finite type star operation *. We study domains D for which the *-class group Cl*(D) equals the t-class group Clt(D) for any finite type star operation *, and we indicate examples of PvMD's D such that Cl*(D)⊊Clt(D). We also compute Clv(D) for certain valuation domains D.