The Moore-Penrose inverse over a commutative ring

Abstract

Let R be a commutative ring with 1 and with an involution a → ā, and let M_R be the category of finite matrices over R with the involution (a_ij) → (a_ij)* = (^-a_ij) A matrix A:m → n in M_R of determinantal rank r such that u(A) = Σ _{α ∈ Q r.m} Σ _{α ∈ Q r.m} det A_αβ -det^--Ā-_αβ 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 M_R, 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.