COMPLEX RINGS AND QUATERNION RINGS

In [4], complex ringsC ( R ; − 1), quaternion ringsH ( R ; − 1 , − 1) and octonion ringsO ( R ; − 1 , − 1) are studied for any ringR . For the real numbers R , C ( R ; − 1) is the complex numbers, H ( R ; − 1 , − 1) is the Hamilton’s quaternionsand O ( R ; − 1 , − 1) is the Cayley-Graves’s octonions. In view of progress of the quaternions, generalized quaternion algebrasa,b F are introduced for commutative fields F and nonzero elements a, b ∈ F , and these quaternion algebrashave been extensively studied as number theory. In this paper, we use H ( F ; a, b ) instead of a,b F . For a division ringD and nonzero elements a, b in the center of D , we introduce generalized complex ringsC ( D ; a ) and generalized quaternion ringsH ( 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 ringand C ( D ; a ) is a simple ringor a direct sumof two simple rings. Main purpose of this paper is to study structures of these simple rings. We also study the case of 2 = 0.