Categories of matrices with only obvious Moore-Penrose inverses

AbstractLet R be an associative ring with 1 and with an involution a → ā, and let MR be the category of finite matrices over R with the involution (aij) → (aij)∗ = (āji). Then the following two statements are equivalent: (i) If A in MR has a Moore-Penrose inverse with respect to ∗, then A is permutationally equivalent to a matrix of the form B000 with B invertible. (ii) If 1 = ∑aā in R, then at most one of the a's is not zero