Note on the Krull dimension of rings of integer-valued polynomials

Let D be an integral domain with quotient field $K,$ E a subset of K and X an indeterminate over K. The set Int(E,D):=∈ K[X];\; f(E)⊆ D\, of integer-valued polynomials on E over D, is known to be an integral domain. The purpose of this note is to calculate the Krull dimension of Int(E,D) across various classes of integral domains D and specific subsets E of D. We further extend our study to the ring IntB(E,D):=∈ B[X];\; f(E)⊆ D\, where B is an integral domain containing D