Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution

Abstract

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1 − ba (resp., ba) being pseudo core invertible while 1 − ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1 − ba (resp., ba) is given in terms of 1 − ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.

Keywords:

• Cline’s formula
• Jacobson’s lemma
• Moore-Penrose inverse
• pseudo core inverse

AMS SUBJECT CLASSIFICATIONS:

• 15A09
• 16W10

Acknowledgment

The authors wish to thank the editor and reviewers sincerely for their constructive comments and suggestions that have improved the quality of the paper.

Funding

This research is supported by the National Natural Science Foundation of China (No. 11771076), the Fundamental Research Funds for the Central Universities, the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX18 0053).