Let R be an associative ring. Drazin proved in 4 that if a and w ∈ R are Drazin invertible such that aw = wa = 0, then a + w is also Drazin invertible. The same results holds for Moore-Penrose inverses in a ring with involution under the condition aw* = a*w = 0. The generalized invertibility of the sum of two elements is very useful and many authors investigated the sum under different conditions. As the inverse along an element is a generalization of the Drazin inverse and the Moore-Penrose inverses, we give some additive results of the inverse along an element of two invertible elements along elements in an associative ring. If a and w are invertible along d and c respectively, then we show under some conditions that a + w is invertible along some t related to d and c. Moreover, we give the expression of the inverse (a + w) ^{||t}. As an application, we study the inverse along an element of a 2 × 2 block matrix. Various examples are given to illustrate our results.
Laghmam et al. (Wed,) studied this question.