Existence of subsets E with IntR(E, D) not equal to Int(D) for D = Z + TQ[T]

Determine whether there exists a subset E of the integral domain D = Z + TQ[T] such that the ring of integer-valued rational functions IntR(E, D) is not equal to the ring of integer-valued polynomials Int(D).

Background

In Example 1.14, the domain D = Z + TQ[T] is a d-ring, implying IntR(D) = Int(D). The authors ask whether for some subsets E of D, the ring of integer-valued rational functions differs from the classical integer-valued polynomial ring.

This investigates whether introducing rational functions evaluated on a subset E of D can enlarge the ring beyond Int(D) even when D is a d-ring.

References

Since the integral domain D in Example 1.14 lies between Z[T] and Q[T], it follows from [3, Lemma VII.2.10] that D is a d-ring, and thus IntR(D) coincides with the classical ring Int(D). However, it remains an open question whether IntR(E, D) # Int(D) for some subsets E of D.

— On the Krull dimension of rings of integer-valued rational functions (2412.07931 - Chems-Eddin et al., 10 Dec 2024) in Remark 1.16(1), Section 1