The strong two-generator property in rings of integer-valued polynomials determined by finite sets

Abstract

Let D be an integral domain and E = {e 1 , . . . , e k } 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 letdenote 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 twogenerated ideal I of a ring R has the property that the first of its two-generators 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 twogenerator 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 (in Theorem 4) that if D is an integral domain and E 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. In certain cases where D is not a Bezout domain and E ⊂ D is nonempty and finite, our results allow us to more closely analyze ideal generation problems in Int(E, D). If an element α of a ring R can be chosen as one of two generators of every two-generated ideal I in which it is contained, then α is called a strong two-generator of R. We close in Corollary 8 by showing that if D is a Dedekind domain which is not a principalMathematics Subject Classification (1991): 13B25, 11S05, 12J10, 13E05, 13G05.

The n-generator property in rings of integer-valued polynomials determined by finite sets

“…. , a r } of D, and let Int(E, D) = {f ∈ Q[X] : f (E) D}, the ring of integer-valued polynomials on D with respect to the subset E. As in [9], we fix polynomials F = [2] and [3], this is called the almost strong Skolem property. In [9] this is called the strong Hilbert property.…”

“…In [3,Theorem 4], Chapman, Loper, and Smith show that, for finite non-empty E, the integral domain D is a Bezout domain if and only if Int(E, D) has the strong 2-generator property. Thus, both D and Int(E, D) must be Prüfer domains in this case.…”

“…(This fact also follows from [8,Theorem 1], where it is shown that every strongly 2-generated ideal is invertible.) In light of [9, Corollary 7], our original motivation for this research was to determine whether or not [3,Theorem 4] could be generalized to the n-generator and strong (n + 1)-generator properties for arbitrary positive integer n, so that iterating the Int(E, D) construction would yield an algebraic construction of a sequence of Prüfer domains whose finitely generated ideals require an ever larger number of generators. As we shall see below (Theorem 3 and Corollary 4), this is not the case.…”

“…The key point is the last paragraph of the proof of [3,Theorem 4], which is a proof of a strong stable range type condition for the ring Q[X]. For our next result, it suffices to note that the ring Q[X] has 2 in the stable range; that is, given polynomials …”

Integer-valued polynomial in valued fields with an application to discrete dynamical systems

“…This case is well described by McQuillan [26] for subsets of any integral domain (see also [8, Exercises IV.1, V.2, VI.20, VIII.25 and VIII.28] and [17]). For finite subsets of a valued field K, one can say a bit more.…”