On the Krull dimension of rings of integer-valued rational functions

Let $D$ be an integral domain with quotient field $K$ and $E$ a subset of $K$. The $\mathit{\text{ring}}\mathit{\text{of}}\mathit{\text{integer-valued}}\mathit{\text{rational}}\mathit{\text{functions}}\mathit{\text{on}}$ $E$ is defined as $${\mathrm{i}\mathrm{n}\mathrm{t}}_R(E,D):=\{\phi \in K(X);\phi (E)\subseteq D\}\mathrm{.}$$ The main goal of this paper is to investigate the Krull dimension of the ring ${\mathrm{i}\mathrm{n}\mathrm{t}}_R(E,D)\mathrm{.}$ Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.