Some results on the integer-valued polynomials over matrix rings

Abstract

ABSTRACTIn this paper, we study the ring of integer-valued polynomials over matrix rings where D is an integral domain with the field of fractions K. We introduce equalizing ideal of Mn(𝔞) in Int(Mn(D)) where 𝔞 is an ideal of D. We show that, if is finite then the set of distinct ideals of the form of Int(Mn(D)) is finite. Also, we prove that . 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 is finite, where 𝔪 is the maximal ideal of D. We present some properties of maximal ideals of Int(Mn(D)). Also, we generalize the Skolem closure of an ideal of Int(Mn(D)). Among the other results, we present some relations between prime ideals of Mn(D) and prime ideals of Int(Mn(D)). Finally, we state a lower bound of Krull dimension of integer-valued polynomials ring Int(Mn(D)).