Prüfer domains of integer-valued polynomials and the two-generator property

Abstract

Let V be a valuation domain and let E be a subset of V. For a rank-one valuation domain V, there is a characterization of when Int(E,V) is a Prüfer domain. For a general valuation domain V, we show that Int(E,V) is a Prüfer domain if and only if E is precompact, or there exists a rank-one prime ideal P of V and Int(E,VP) is a Prüfer domain. Then we show that the following statements are equivalent: (1) Int(E,V) is a Prüfer domain; (2) it has the strong 2-generator property; (3) it has the almost strong Skolem property. In this case, by showing that Int(E,V) is almost local-global, we obtain that it has the stacked bases property and the Steinitz property. For a Prüfer domain D, we show that the following statements are equivalent: (1) Int(D) is a Prüfer domain; (2) it has the 2-generator property; (3) it has the almost strong Skolem property. In this case, Int(D) is not necessarily almost local-global, but we show that it has the Steinitz property.