Some Boundedness Conditions for Rings of Integer-Valued Polynomials

Abstract

Let R be an integral domain with quotient field K, and let Int(R) be the ring of integer-valued polynomials on R. That is Int(R) = {f ∈ K[X] | f(R) ⊆ R}. It is known that if R is an almost Dedekind domain, then the double boundedness condition, which characterizes when Int(R) is a Prüfer domain, is equivalent to another double boundedness condition which arises in characterizing when Int(R) ≠ R[X]. After recasting the first of these conditions so that it applies to any integral domain R, instead of just to Prüfer domains, it is shown that these two double boundedness conditions are equivalent if R is a finite conductor domain, but not for all integral domains R.