Integer-valued polynomials on subsets of upper triangular matrix rings

Abstract

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring ${\text{Int}}_{T_n(K)}(T_n(D)):=\{f\in T_n(K)[x]|f(T_n(D))\subseteq T_n(D)\}$ is a ring, where D is an integral domain with field of fractions K. Let $R_1\subseteq R_2$ be commutative rings with identity. In this paper, we study the set ${\text{Int}}_{T_n(R_2)}(\Omega ,T_n(R_1)):=\{f\in T_n(R_2)[x]|f(\Omega )\subseteq T_n(R_1)\}$ for some subsets $\Omega \subseteq T_n(R_1)$. We generalize Frisch’s result and show that ${\text{Int}}_{T_n(R_2)}(T_n(R_1)):={\text{Int}}_{T_n(R_2)}(T_n(R_1),T_n(R_1))$ is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.

KEYWORDS:

• Integer-valued polynomial
• upper triangular matrix rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F20
• 16D25
• 16S50

Acknowledgments

We would like to thank the referee for valuable comments and suggestions that helped to improve the paper.