Integer-Valued Polynomials over Matrix Rings

Abstract

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 non-Noetherian.