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