In this paper, we study the ring of integer-valued polynomials Int (M_n (O _K)):={f ∈ M_n (K)
x~|~ f (M_n (O _K)) ⊆ M_n (O _K)\} Int (M n (OK)):= f∈ M n (K) x| f (M n (OK))⊆ M n (OK),
where K is a number field and O _K OK is the ring of algebraic integers of K. We show that
for a prime number p ∈ Z p∈ Z, the polynomial f_ p, n (x):=(x^ p^ nx)(x^ p^ n-1-x) ... (x^ px) p
fp, n (x):=(xpn-x)(xpn-1-x)…(xp-x) p is an element of Int (M_n (O _K)) Int (M n (OK)) if and
only if p is a totally split prime in O _K OK. Also, we consider the ring Int _ M_n (Q)(M_n (O …

When D is a commutative integral domain with field of fractions K, the ring Int (D)={f∈ K [x]| f
(D)⊆ D} of integer-valued polynomials over D is well-understood. This article considers the
construction of integer-valued polynomials over matrix rings with entries in an integral
domain. Given an integral domain D with field of fractions K, we define Int (M n (D)):={f∈ M n
(K)[x]| f (M n (D))⊆ M n (D)}. We prove that Int (M n (D)) is a ring and investigate its structure
and ideals. We also derive a generating set for Int (M n (ℤ)) and prove that Int (M n (ℤ)) is …