| In the literature, only the Clifford algebra over a field F is defined and considered. The theory of such Clifford algebra is very important in representation theory of groups, theory of quadratic forms, statistical mechanics and quantum field theory. In this paper, the Clifford R-algebra over a commutative ring R is defined. In addition, we proved uniqueness theorem and existence theorem of Clifford R-algebra. We also defined R-generalized quaternions algebra and R-generalized complex algebra, and proved that they are Clifford R-algebra. It is easy to see that exterior algebra, quaternions algebra, complex(field) are Clifford R-algebra from this paper. |
