Integer-valued Polynomials over Quaternion Rings
When D is an integral domain with field of fractions K, the ring Int(D) of integer-valued polynomials over D is defined to be the set of all polynomials f(a) in K[x] such that f(a) is in D for all a in D. The goal of this dissertation is to extend the integer-valued polynomial construction to certain noncommutative rings. This is accomplished by considering a family of quaternion algebras over the integers. When R is such an algebra, we provide a definition for the set Int(R) of integer-valued polynomials over R, prove that Int(R) has a ring structure, and investigate elements, generating sets, and prime ideals of Int(R). We also consider the related concept of integer-valued polynomials on subsets of these quaternion algebras.