Some remarks on Prufer ☆-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:= {T-lambda|lambda is an element of Lambda} 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 (c(D)(fg))* = (c(D)(f)c(D)(g))* (respectively, (cD (fg))(w) = (c(D)(f)c(D)(g))(w)) for all 0 not equal f, g is an element of 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 Cl-t (D) for any finite type star operation *, and we indicate examples of PvMD's D such that Cl*(D) not subset of Cl-t(D). We also compute Cl-v(D) for certain valuation domains D. (C) 2007 Elsevier Inc. All rights reserved.