Generalized Inverses and Units in a Unitary Ring

Let R be a unitary ring and a,b ∈ R with ab = 0. We find the 2/3 property of Drazin invertibility: if any two of a, b and a+b are Drazin invertible, then so is the third one. Then, we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units. As applications, the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism, and the existence and expression of the inverse along a product of two regular elements are obtained, which generalizes the main result of Mary and Patrício (2016) about the group inverse of a product.