Rings of integer-valued rational functions Original Research Article

Let D be an integral domain which differs from its quotient field K. The ring of integer-valued rational functions of D on a subset E of D is defined as IntR(E, D) = f(X) set membership, variant K(X)f(E) subset of-. We write IntR(D) for IntR(D, D). It is easy to see that IntR(D) is strictly larger than the more familiar ring Int(D) of integer-valued polynomials precisely when there exists a polynomial f(X) set membership, variant D[X] such that f(d) is a unit in D for each d set membership, variant D. In fact, there arc striking differences between IntR(D) and Int(D) in many of the cases where they are not equal. Rings of integer-valued rational functions have been studied in at least two previous papers. The purpose of this note is to consolidate and greatly expand the results of these papers. Among the topics included, we give conditions so that IntR(E, D) is a Prüfer domain, we study the value ideals of IntR(E, D) (for example, we show that IntR(K, D) satisfies the strong Skolem property provided it is a Prüfer domain), and we study the prime ideals of IntR(E, D) (for example, we show that if V is a valuation domain, then each prime ideal of IntR(V) above the maximal ideal m of V is maximal if and only if m is principal).