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

Abstract

Let D be an integral domain with quotient field K and E a subset of K. The ring of integer-valued rational functions on E is defined as

$$\begin{aligned} \mathrm {Int^R}(E,D):=\lbrace \varphi \in K(X);\; \varphi (E)\subseteq D\rbrace . \end{aligned}$$

The main goal of this paper is to investigate the Krull dimension of the ring \(\mathrm {Int^R}(E,D).\) Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.