When and are Prüfer domains

Let D be an integral domain, X be an indeterminate over D, and be the power series ring over D. For , let cD(f) denote the ideal of D generated by the coefficients of f. Let , , , and . We show that D is a Krull domain if and only if is a Prüfer domain, if and only if is a valuation domain for each maximal t-ideal P of D, if and only if is a PvMD in which each t-ideal is divisorial. We also show that D is a Dedekind domain if and only if is a Prüfer domain, if and only if is a valuation domain for each maximal ideal M of D.