Unitary Elements in Simple Artinian Rings Original Research Article

The problem of determining when a unitary element is a product of Cayley unitary elements is completely solved for simple artinian rings of characteristic not 2. Theorem 1. Let D be a division ring of characteristic not 2. Suppose that R = Dn assumes an involution which induces a non-identity involution on D. Then any unitary element in R is a product of two Cayley unitary elements. Theorem 2. Let F be a field of characteristic not 2. Suppose that R = Fn assumes an involution * of the first kind. Then any unitary element in R which is a product of Cayley unitary elements must have determinant 1. Conversely, any unitary element in R of determinant 1 is a product of two Cayley unitary elements, except when F = GH(3), n = 2, and * is given by [formula].