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

Abstract

Let R be an integral domain with quotient field K and let Int(R) = { f ε K[X]| f(R) ⊆ R}. In this note we determine when Int(R) = R[X] for an arbitrary integral domain R. More generally we determine when Int(R) ⊆ 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(R S ) = Int(R) 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.