Integer-valued Polynomials over Quaternion Rings

When D is an integral domain with field of fractions K, the ring Int(D) = {f(x) ∈ K[x] | f(D) ⊆ D} of integer-valued polynomials over D has been extensively studied. We will extend the integer-valued polynomial construction to certain noncommutative rings. Specifically, let i, j, and k be the standard quaternion units satisfying the relations i = j = −1 and ij = k = −ji, and define ZQ := {a+bi+cj+dk | a, b, c, d ∈ Z}. Then, ZQ is a noncommutative ring that lives inside the division ring QQ := {a+bi+cj+dk | a, b, c, d ∈ Q}. For any ring R such that ZQ ⊆ R ⊆ QQ, we define the set of integer-valued polynomials over R to be Int(R) := {f(x) ∈ QQ[x] | f(R) ⊆ R}. We will demonstrate that Int(R) is a ring, discuss how to generate some elements of Int(ZQ), prove that Int(ZQ) is non-Noetherian, and describe some of the prime ideals of Int(ZQ).