Generalized inverses of elements and their polarities in rings

Let R be an associative ring with unity 1. The main contribution of this paper is to introduce the notion of generalized quasipolar elements in R as an extension of quasipolar elements of Koliha and Patricio. Several necessary and sufficient conditions of $$a\in R$$ to be generalized quasipolar are derived. Then, we define a class of outer generalized inverses, called weakly generalized Drazin inverses generalizing Koliha’s generalized Drazin inverses. It is shown that $$a\in R$$ has a weakly generalized Drazin inverse if and only if it is generalized quasipolar. Finally, existence criteria for weakly generalized Drazin inverses are obtained.