Prüfer domains of integer-valued polynomials on a subset

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Let D be an integral domain with quotient field K and let E be an additive subgroup of D. We investigate the question of when Int(E,D)={f∈K[X]|f(E)⊆D} is a Prüfer domain. We show that if D is a Krull-type Prüfer domain, an SFT Prüfer domain, or the ring of entire functions, then Int(E,D) is a Prüfer domain if and only if sup{|E/IDM||M∈Max(D)}<∞ for each nonzero fractional ideal I of D. In particular, for a valuation domain D, we also show that: Int(E,D) is a Prüfer domain if and only if Int(E,D) has the strong 2-generator property if and only if Int(E,D) is dense in C(Eˆ,Dˆ) for the uniform convergence topology if and only if E is a precompact subset of K. This gives a partial answer to the question posed by Cahen, Chabert, and Loper.