Some remarks on Prüfer ⋆ --multiplication domains and class groups

الأصل · EN

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 (resp., PvMD) if and only if ((fg))=((f)(g)) (resp., ((fg))ʷ=((f)(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 PvMD's which are not P MD's for a certain finite type star operation. We study domains D for which the --class group (D) equals the t--class group ᵗ(D) for any finite type star operation, and we indicate examples of PvMD's D such that (D) ᵗ(D). We also compute ᵛ(D) for certain valuation domains D.