The generalized inverse of a sum with radical element: Applications Original Research Article

An expression for the generalized inverse of a sum with radical element in a ring with unity is generalized to the case of a sum phi + η, in an additive category image, of a morphism phi with a reflexive Von Neumann regular inverse phi(1,2) and a morphism η of image which is such that 1X + phi(1,2)η is invertible. Also, if image is an additive category with an involution *, phi a morphism with Moore-Penrose inverse phi†, and η such that 1X + phi†η, 1X − λ = 1X − (1X + phi†η)−1(1X − phi†phi)η*phi†*(1X + η*phi†*)−1, and 1Y − μ = 1Y − (1Y + phi†*η*)−1phi†*η*(1Y − phiphi†)(1Y + ηphi†)−1 are invertible, then phi + η − (1Y − phiphi†)η(1X + phi†η)−1(1X − phi†phi) has a Moore-Penrose inverse, given by (1X − λ)−1(1X + phi†η)− phi†(1Y − μ)−1. Relations with results of Wynn, Roth, and Nashed are discussed.