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 Patrício. Several necessary and sufficient conditions of a∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in R$$\end{document} 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∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in R$$\end{document} has a weakly generalized Drazin inverse if and only if it is generalized quasipolar. Finally, existence criteria for weakly generalized Drazin inverses are obtained.