Converse for infinite Krull dimension of IntR(E, D)

Determine whether an infinite Krull dimension of the ring of integer-valued rational functions IntR(E, D) implies that the integral domain D has infinite Krull dimension, for any integral domain D with quotient field K and any nonempty subset E of K.

Background

Proposition 1.1 establishes that dim(IntR(E, D)) ≥ dim(D), and consequently if dim(D) is infinite then dim(IntR(E, D)) is infinite. The authors note that the reverse implication is not known.

This question asks whether the property of having infinite Krull dimension for the ring of integer-valued rational functions necessarily forces the base domain to have infinite Krull dimension, independent of any additional hypotheses on E.

References

From the previous proposition, we deduce that if the dimension of D is infinite, then so is the dimension of IntR(E, D). However, we do not know whether the converse is true or not.

— On the Krull dimension of rings of integer-valued rational functions (2412.07931 - Chems-Eddin et al., 10 Dec 2024) in Remark 1.2, Section 1