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 be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by . We show that D is a P*MD (respectively, PvMD) if and only if (cD(fg))*=(cD(f)cD(g))* (respectively, (cD(fg))w=(cD(f)cD(g))w) 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 PvMDʹs which are not P*MDʹs 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 PvMDʹs D such that Cl*(D) Clt(D). We also compute Clv(D) for certain valuation domains D.