Integer-valued polynomials on subsets of upper triangular matrix rings

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring IntTn(K)(Tn(D)):={f∈Tn(K)[x]|f(Tn(D))⊆Tn(D)} is a ring, where D is an integral domain with field of fractions K. Let R1⊆R2 be commutative rings with identity. In this paper, we study the set IntTn(R2)(Ω,Tn(R1)):={f∈Tn(R2)[x]|f(Ω)⊆Tn(R1)} for some subsets Ω⊆Tn(R1). We generalize Frisch's result and show that IntTn(R2)(Tn(R1)):=IntTn(R2)(Tn(R1),Tn(R1)) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.