The Strong Two-generator Property in Rings of Integer-valued Polynomials Determined by Finite Sets

Let D be an integral domain and E = {e1, . . . , ek} a finite nonempty subset of D. Then Int(E,D) has the strong two-generator property if and only if D is a Bezout domain. If D is a Dedekind domain which is not a principal ideal domain, then we characterize which elements of Int(E, D) are strong two-generators. Let D be an integral domain with quotient field K and E ⊆ D a subset of D. We let Int(E,D) = {f(x) ∈ K[X] | f(x) ∈ D for every x ∈ E} denote the much studied 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)). When D is a Dedekind domain with finite residue fields, the ideal theory of Int(E,D) has generated a considerable amount of attention in the recent mathematical literature. In [3], Gilmer and Smith showed that the finitely generated ideals of Int(Z) satisfy the two-generator property. This result was extended to Int(E,D) where D is a Dedekind domain with finite residue fields and E is a “D-fractional” subset of K (i.e., there exists a d ∈ D such that dE ⊆ D) by McQuillan [7, Theorem 5.5]. In a later paper [4], Gilmer and Smith showed the existence of finitely generated ideals in Int(Z) where the first of the two required generators cannot be chosen at random. If a two-generated ideal I of a ring R has the property that the first of its twogenerators can be chosen at random from the nonzero elements of I, then I is called strongly two-generated. A ring in which each two-generated ideal is strongly two-generated is said to have the strong two-generator property. Under the assumption that a ring has zero Jacobson radical, this property is equivalent to the 1 1/2-generator property [5]. Hence, Int(Z) does not have the strong two-generator property. In this note, we prove the following theorem. 1991 Mathematics Subject Classification. 13B25, 11S05, 12J10, 13E05, 13G05.