Rings of integer-valued rational functions
For \(D\) an integral domain with quotient field \(K\) and \(E \subseteq D\), let \(\text{Int} (E, D)= \{f \in K[x]\mid f(E) \subseteq D\}\) and \(\text{Int}^R (E, D) = \{f \in K (x) \mid f(E) \subseteq D\}\). For a large class of rings, called \(D\)-rings (which includes \({\mathbb Z}\)), \(\text{Int}^R (D, D) = \text{Int} (D, D)\). This paper is devoted to studying \(\text{Int}^R(E, D)\) for non-\(D\)-rings. Conditions are given which insure that \(\text{Int}^R(E, D)\) is Prüfer or Bezout, and the value ideals and prime ideals are studied. Of particular interest is the case \(D = V\), a valuation domain. It is shown that if the maximal ideal, \(M\), of \(V\) is principal, then all prime ideals of \(\text{Int}^R(E, V)\) over \(M\) are maximal, while if \(M\) is not principal this fails for \(\text{Int}^R(V, V)\).
scientific article; zbMATH DE number 1308233