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

Let Vbe a valuation domain and let Ebe a subset of V. For a rank-one valuation domain V, there is a characterization of when Int(E, V) is a Prufer domain. For a general valuation domain V, we show that Int(E, V) is a Prufer domain if and only if Eis precompact, or there exists a rank-one prime ideal Pof Vand Int(E, V-P) is a Prufer domain. Then we show that the following statements are equivalent: (1) Int(E, V) is a Prufer 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 Prufer domain D, we show that the following statements are equivalent: (1) Int(D) is a Prufer 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. (C) 2021 Elsevier Inc. All rights reserved.