Integer-valued polynomials over subsets of matrix rings

Abstract

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 , then we prove that is a ring. We show that is not Noetherian where I is an ideal of . We present a direct proof to show that the set is a ring where is the upper triangular matrix ring over D and E is a subset of D containing 0. We find out that, if is not Noetherian then is a non-Noetherian ring. Also, we state a lower bound for Krull dimension of integer-valued polynomials ring . We further introduce , where and we demonstrate that it is a ring. Finally, we illustrate that if is a Noetherian ring then is Noetherian, where E is a subset of D containing 0.