The characterizations of one-sided generalized inverses

Abstract

In this article, some characterizations of one-sided generalized inverses are investigated. Let R be a *-ring and $a\in R.$ It is shown that for $k\ge 2,$ a* is right $(a^k,a)$ invertible if and only if $a\in a^{2}R\cap R^{†}.$ In particular, the expression of a^† are given whenever $a\in a^{2}R\cap R^{†}.$ Moreover, it is also proven that $a\in a^{2}R\cap R^{†}$ if and only if ${(a^{\ast }a)}^n$ is right invertible along a if and only if $a^{\ast }{a}^n$ is right invertible along a, where n is an arbitrary positive integer. Finally, we present some characterizations of $a\in a^{2}R\cap R^{†}$ by using projections and units.

Keywords:

• Core inverse
• inverse along an element
• rings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A09
• 16W10

Acknowledgments

The authors are grateful and express their sincere thanks to the editor and the referee for their valuable and detailed comments on the manuscript. In particular, Remarks 2.9 and 2.12 and Lemmas 3.1 and 3.3 were suggested to the authors by the referee.

Additional information

Funding

The research is supported by the NSF of China (11771076), NSF of Jiangsu Province (BK20170589), NSF of Jiangsu Higher Education Institutions of China (15KJB110021), and China Postdoctoral Science Foundation (2017M611920).