Unitary elements in simple Artinian rings (Q1902135)

Let \(R\) be a ring with involution *. An element \(u \in R\) is unitary if \(uu^*=1=u^*u\), and is a Cayley unitary if \(u=(1-k)(1+k)^{-1}\) for some \(k^*=-k\in R\). The authors characterize those unitary elements in simple Artinian rings which are products of Cayley unitaries. The first theorem shows that when \(R=M_n(D)\) for \(D\) a division ring, \(\text{char }D\neq 2\), and * is not the identity on \(D\cdot I_n\), then every unitary in \(R\) is a product of two Cayley unitary elements. If \(^*\) is the identity on \(D\cdot I_n\), then \(D=F\), a field, and * is said to be of the first kind. In this situation every Cayley unitary has determinant one. The second theorem is that if \(R=M_n(F)\) with \(\text{char }F \neq 2\) and * is of the first kind, then every unitary element in \(R\) of determinant one is a product of two Cayley unitary elements, except when \(R=M_2(\mathbb{F}_3)\) and \(A^*=\text{Diag}(1,- 1)A^t\text{Diag}(1,-1)\), where \(A^t\) is the usual transpose of \(A\) in \(R\). This exceptional case has \(I_2\) as the only Cayley unitary.