Hurwitz Split Quaternions

Abstract

In this work, we introduce the set of Hurwitz split quaternions, , which is an extension of the split quaternions over the integers, . In comparison to , the Hurwitz splits only have integer or half-integer coefficients, not both. We demonstrate that it has a ring structure and examine certain features of this notion by pointing out the differences from . We also provide the matrix representation of the ring of Hurwitz split quaternions. In addition, we study some of its prime ideals and show that this ring is Noetherian. Then we describe the set of integer-valued polynomials over to be . Once we prove forms a ring, we investigate some essential properties of this ring.