The conditions Int(R)⊆RS[X] and Int(RS)=Int(R)S for integer-valued polynomials

Let R be an integral domain with quotient field K and let Int(R) = {f is an element of K[X] \ f(R) subset of or equal to R}. In this note we determine when Int(R) = R[X] for an arbitrary integral domain R. More generally we determine when Int(R) subset of or equal to R-S[X] for a multiplicative subset S of R. In the case that R is an almost Dedekind domain with finite residue fields we also determine when Int(Rs)= Int(li)s for each multiplicative subset S of R, and show that if this holds then finitely generated ideals of Int(R) can be generated by two elements. (C) 1998 Elsevier Science B.V.