# COMPLEX RINGS AND QUATERNION RINGS

### Abstract

In [4], complex rings C(R;−1), quaternion rings H(R;−1,−1) and octonion rings O(R;−1,−1) are studied for any ring R. For the real numbers R, C(R;−1) is the complex numbers, H(R;−1,−1) is the Hamilton’s quaternions and O(R;−1,−1) is the Cayley-Graves’s octonions. In view of progress of the quaternions, generalized quaternion algebras a,b F areintroduced for commutative ﬁelds F and nonzero elements a,b ∈ F, and these quaternion algebras have been extensively studied as number theory. In this paper, we use H(F;a,b) instead of a,b F .For a division ring D and nonzero elements a,b in the center of D, we introduce generalized complex rings C(D;a) and generalized quaternion rings H(D;a,b), and study the structure of these rings. We show that, if 2 = 0, that is, the characteristic of D is not 2, then H(D;a,b) is a simple ring and C(D;a) is a simple ring or a direct sum of two simple rings. Main purpose of this paper is to study structures of these simple rings. We also study the case of 2 = 0.