On the flatness of Int(D) as a D[X]-module

Abstract

Let D be an integral domain with quotient field K and X an indeterminate. We show that if D is either Krull or Noetherian, then Int(D) := {f ∈ K[X] : f(D) ⊆ D} is flat over D[X] if and only if Int(D) = D[X]. Then, we give several examples of domains D with Int(D) not flat over D[X]. Also, we generalize our investigations to the case of Int(E,D) := {f ∈ K[X] : f(E) ⊆ D}, where E is a subset of D.