Robert Gilmer’s Contributions to the Theory of Integer-Valued Polynomials

denote the ring of integer-valued polynomials on D with respect to the subset E (for ease of notation, if E = D, then set Int(D,D) = Int(D)). Gilmer’s work in this area (with the assistance of various co-authors) was truly groundbreaking and led to numerous extensions and generalizations by authors such as J. L. Chabert, P. J. Cahen, D. McQuillan and A. Loper. In this paper, we will review Gilmer’s papers dedicated to this subject. We close with an elementary analysis of polynomial closure in integral domains, a topic which Gilmer motivated with a characterization of which subsets S of Z define the ring Int(Z) in [18]. Before proceeding, please note that we use Q to represent the rationals, Z the integers, N the natural numbers and P the primes in Z. It is clear that Gilmer’s interest in the rings Int(E,D) was motivated by his early work on multiplicative ideal theory and the theory of Prüfer domains. In particular, there was a problem open in the early 60’s regarding the number of required generators for a finitely generated ideal of a Prüfer domain. It was at the time well known in the Noetherian case (i.e., for a Dedekind domain) that every ideal could be generated by two elements, one of which could be chosen to be an arbitrary non-zero element of the ideal.