RINGS OF INTEGER-VALUED RATIONAL FUNCTIONS

Let D be an integral domain which differs from its quotient field K. T he ring of integer-valued rational functions of D on a subset E of D i s defined as Int(R)(E, D) = {f(X) is an element of K(X)\f(E) subset of or equal to D}. We write Int(R)(D) for Int(R)(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)E D[X] such that f(d) is a unit in D for each d is an element of D. In fact, there are striking differences between Int(R)(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 res ults of these papers. Among the topics included, we give conditions so that Int(R)(E,D) is a Prufer domain, we study the value ideals of Int (R)(E,D) (for example, we show that Int(R)(K,D) satisfies the strong S kolem property provided it is a Prufer domain), and we study the prime ideals of Int(R)(E,D) (for example, we show that if V is a valuation domain, then each prime ideal of Int(R)(V) above the maximal ideal m o f V is maximal if and only if m is principal). (C) 1998 Elsevier Scien ce B.V. All rights reserved.