On a question of Drazin

Abstract

Let R be an associative ring. In this paper, we give a positive answer to the question of M.P. Drazin, “whether $a\in R$ has a unique core-quasi-nilpotent decomposition implies that it must be quasi-invertible”.

Keywords:

• Core-quasi-nilpotent decompositions
• Drazin inverses
• quasi-inverses

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A09
• 06A11

Acknowledgments

The authors are highly grateful to the referee for his/her valuable comments and suggestions which led to improvements of this paper. In particular, the main result in Section 3 is suggested to the authors by the referee.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [No. 11801124], China Postdoctoral Science Foundation [No. 2020M671068], and the Natural Science Foundation of Anhui Province [No. 1808085QA16].