Categories of matrices with only obvious Moore-Penrose inverses (Q1095987)

Let R be an associative ring with 1 and with an involution \(a\to \bar a\), and let \(M_ R\) be the category of finite matrices over R with the involution \((a_{ij})\to (a_{ij})^*=(\bar a_{ji})\). The authors prove that the following two statements are equivalent: (1) If A in \(M_ R\) has a Moore-Penrose inverse with respect to *, then A is permutationally equivalent to a matrix of the form \(\left( \begin{matrix} B\quad 0\\ 0\quad 0\end{matrix} \right)\) with B invertible. (2) If \(1=\Sigma a\bar a\) in R, then at most one of the a's is not zero. Some other equivalent conditions are also contained.