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

Abstract

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 Prufer domain. For a general valuation domain V, we show that Int ( E , V ) is a Prufer domain if and only if E is precompact, or there exists a rank-one prime ideal P of V and 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.