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 non-commutative rings. Specifically, let i, j, and k be the standard quaternion units satisfying the relations i2=j2=−1 and ij=k=−ji, and define ZQ:={a+bi+cj+dk|a,b,c,d∈Z}. Then, ZQ is a non-commutative 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).