A classification of all is a Prüfer domain

Let be an integral domain with quotient field . The ring of integer-valued polynomials over is defined by . It is known that if is a Prüfer domain, then is an almost Dedekind domain with all residue fields finite. This condition is necessary and sufficient if is Noetherian, but has been shown to not be sufficient if is not Noetherian. Several authors have come close to a complete characterization by imposing bounds on orders of residue fields of and on normalized values of particular elements of . In this note we give a double-boundedness condition which provides a complete charaterization of all integral domains such that is a Prüfer domain.