The Moore-Penrose inverse over a commutative ring

AbstractLet R be a commutative 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). A matrix A:m → n in MR of determinantal rank r such that u(A) = ∑α∈Qr.m∑α∈Qr.m det Aαβ det Aαβ has a Moore-Penrose inverse u(A)† in R is said to be Moore invertible with Moore idempotent u(A)u(A)† if u(A)u(A)†A = A. For every matrix A of MR, A has a Moore-Penrose inverse with respect to ∗ if and only if A is the sum of Moore invertible matrices whose Moore idempotents are pairwise orthogonal