Formally Integrally Closed Domains and the RingsR((X)) andR{{X}}

LetRbe an integral domain. Forf R [ X ] letAfbe the ideal ofRgenerated by the coefficients off. We defineRto be formally integrally closed (Afg)t = (AfAg)tfor all nonzerof, g R [ X ] . Examples of formally integrally closed domains include locally finite intersections of one-dimensional Prüfer domains (e.g., Krull domains and one-dimensional Prüfer domains). We study the ringsR((X)) = R [ X ] NandR{{X}} = R [ X ] NtwhereN = {f R [ X ] Af = R} andNt = {f R [ X ] (Af)t = R}. We show thatRis a Krull domain (resp., Dedekind domain) R{{X}} (resp.,R((X))) is a Krull domain (resp., Dedekind domain) R{{X}} (resp.,R((X))) is a Euclidean domain every (principal) ideal ofR{{X}} (resp.,R((X))) is extended fromR Ris formally integrally closed and every prime ideal ofR{{X}} (resp.,R((X))) is extended fromR.