Generalized inverses of a sum of morphisms Original Research Article

Let image be an additive category. Suppose that phi:X→X is a morphism of image with group inverse phi#, and η:X→X is a morphism of image such that 1X+phi#η is invertible. Let var epsilon=(1X−phiphi#)ηα(1X−phi#phi), γ=α(1X−phi#phi)ηphi#β,δ=αphi#η(1X−phiphi#)β, where α=(1X+phi#η)−1,β=(1X+ηphi#)−1. Then f=phi+η−var epsilon has a group inverse if and only if 1X−γ,1X−δ are invertible. Similar result for MP-inverse is also given, which completed Huylebrouckʹs result.