Characterizations for pseudo core inverses in a ring with involution

Abstract

Let R be a unital $\ast$-ring. In this paper, we give several characterizations for the pseudo core and dual pseudo core inverses of an element by units in R. Moreover, their relations with $\ast$-DMP are obtained. As a special case, the existence criteria for EP elements ($\ast$-DMP with index 1) are given. Then, the double commutant property of hybrid (b, c)-inverses is considered. Finally, the relations of the inverse along an element and the hybrid (b, c)-inverse, the (dual) pseudo core inverses and the hybrid (b, c)-inverse are investigated.

Keywords:

• Pseudo core inverses
• Drazin inverses
• Moore–Penrose inverses
• $\ast$-DMP
• hybrid (b
•  c)-inverses
• double commutant property

AMS Subject Classifications:

• 15A09
• 16U99

Acknowledgements

The authors are highly grateful to the referee for his/her valuable comments and suggestions which greatly improved this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the Fundamental Research Funds for the Central Universities, the Natural Science Foundation of Anhui Province [No. 1808085QA16] and the Portuguese Funds through FCT – ‘Fundação para a Ciência e a Tecnologia’ [project number UID-MAT-00013/2013].