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.