Additive results for the inverse along an element in a ring

Abstract

Let $\mathcal{R}$ be an associative ring. Drazin proved in \cite{Drazin1} that if $a$ and $w\in\mathcal{R}$ 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 $2\times 2$ block matrix. Various examples are given to illustrate our results.