Integer-valued polynomials over subsets of matrix rings

In this article, we study the ring of integer-valued polynomials over some matrix rings. Let D be an integral domain with the field of fractions K and I be an ideal of D. We introduce Int(M_n(I), M_n(D)) = {f ∈ M_n(K)[x] | f(M_n(I)) ⊆ M_n(D)}, then we prove that (M_n(I), M_n(D) is a ring. We show that Int(M_n(I), M_n(Z) is not Noetherian where I is an ideal of Z. We present a direct proof to show that the set Int(T_n(E); T_n(D)) = {f ∈ T_n (K)[x] | f(T_n(E)) ⊆ T_n(D)} is a ring where T_n(D) is the upper triangular matrix ring over D and E is a subset of D containing 0. We find out that, if Int(E,D) is not Noetherian then Int(T_n(E), T_n(D)) is a non-Noetherian ring. Also, we state a lower bound for Krull dimension of integer-valued polynomials ring Int(T_n(E), T_n(D)). We further introduce Int(H_n(I), H_n(D)) = {f ∈ H_n(K)[x] | f (H_n(I)) ⊆ H_n(D)}, where H_n(D) = f[a_ij] ∈ T_n(D) | a₁₁ = a₂₂ = … = a_nn} and we demonstrate that it is a ring. Finally, we illustrate that if Int(H_n(E), H_n(D)) is a Noetherian ring then Int(E,D) is Noetherian, where E is a subset of D containing 0.