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

Abstract

Let D be an integral domain and E a non-empty finite subset of D. For n ≧ 2, we show that D has the n-generator property if and only if Int(E, D) has the n-generator property if and only if Int(E, D) has the strong (n + 1)-generator property. Thus, iterating the Int(E, D) construction cannot produce Prufer domains whose finitely generated ideals require an ever larger number of generators. We also show that, for n ≧ 2, a non-zero polynomial f ∈Int(E, D) is a strong n-generator in Int(E, D) if and only if f (a) is a strong n-generator in D for all a ∈E.