On rings of integer-valued rational functions

Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of $D\mathit{\text{-valued}}B\mathit{\text{-rational}}\mathit{\text{functions}}\mathit{\text{on}}E$, denoted by $In{t}_B^R(E,D)$, which naturally extends the concepts of integer-valued polynomials, defined as $In{t}_B^R(E,D)=\{f\in B(X);f(E)\subseteq D\}\mathrm{.}$ The notion of $In{t}_B^R(E,D)$ boils down to the usual notion of integer-valued rational functions when the subset $E$ is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from $In{t}^R(E,D)$ to $D$.