When D((X)) and D{{X}} are Prüfer domains

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