Some results on the integer-valued polynomials over matrix rings

In this paper, we study the ring of integer-valued polynomials over matrix rings Int(M-n(D)) {f is an element of M-n(K)[x] vertical bar f(M-n(D)) subset of M-n(D)] where D is an integral domain with the field of fractions K. We introduce equalizing ideal qM(n(alpha)) of M-n(alpha) in Int(M-n(D)) where alpha an ideal of D. We show that, if M-n(D)/qM(n)(alpha) is finite then the set of distinct ideals of the form J(Mn)(alpha)(,A) of Int(M-n(D)) is finite. Also, we prove that M-n(q(alpha)) subset of q(Mn)(alpha). Using this inclusion we obtain that, if D is a Noetherian local one-dimensional domain with finite residue field, which is not unibranched, then the set of ideals of the form J(Mn)(m)(,A) is finite, where is the maximal ideal of D. We present some properties of maximal ideals of Int(M-n(D)). Also, we generalize the Skolem closure of an ideal of Int(M-n(D))). Among the other results, we present some relations between prime ideals of M-n(D) and prime ideals of Int(M-n(D)). Finally, we state a lower bound of Krull dimension of integer-valued polynomials ring Int(M-n(D))