The inverse along a product and its applications

Abstract

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix.

Keywords:

• von Neumann regularity
• inverse along an element
• Green’s relations
• matrices over a ring

AMS Subject Classifications:

• 15A09
• 16E50

Acknowledgements

The authors are highly grateful to the referee for valuable comments which led to improvements of the paper. In particular, Remarks 3.2 and 3.4 were suggested to the authors by the referee. The first author is grateful to China Scholarship Council for supporting him to purse his further study in University of Minho, Portugal.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Pedro Patrício and Yulin Zhang were financed by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the ‘Fundação para a Ciência e a Tecnologia’, through the Project [PEst-OE/MAT/UI0013/2014]; Jianlong Chen and Huihui Zhu were supported by the National Natural Science Foundation of China [grant number 11201063], [grant number 11371089]; the Specialized Research Fund for the Doctoral Program of Higher Education [grant number 20120092110020]; the Natural Science Foundation of Jiangsu Province [grant number BK20141327]; the Foundation of Graduate Innovation Program of Jiangsu Province [grant number CXLX13-072]; the Scientific Research Foundation of Graduate School of Southeast University and the Fundamental Research Funds for the Central Universities [grant number 22420135011].