The complete number ring and power series

Abstract

This article intends to provide a basic study for the complex quaternion ring . We give a precise description of the relations between the norm of the product of two elements in and the product of the norms of the two elements, and develop a new powerful method, namely, the local direct sum decomposition method, to study the ring . Using the “natural basis” introduced by us, we extend the classical rules for matrix calculus to . We also determine the “mixed spherical shell” within which the power series is convergent and without which the power series is divergent.