Some remarks on Prüfer ⋆-multiplication domains and class groups

Abstract

Let D be an integral domain with quotient field K and let X be an indeterminate over D. Also, let T : = { T λ | 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, P vMD) if and only if ( c D ( f g ) ) * = ( c D ( f ) c D ( g ) ) * (respectively, ( c D ( f g ) ) w = ( c D ( f ) c D ( 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 P vMD'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 P vMD's D such that Cl * ( D ) ⊊ Cl t ( D ) . We also compute Cl v ( D ) for certain valuation domains D.