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Abstract
In this article, we investigate the Drazin invertibility for the elements of an arbitrary semiring. We give necessary and sufficient conditions for the existence and expressions of the Drazin inverse of an element in an arbitrary semiring. Moreover, we consider the product paq under some additional necessary conditions for which the Drazin inverse of the product paq exists.
Acknowledgements
The authors are grateful to the handling editor and anonymous referees for their careful review and valuable suggestions. This research was supported by the grants from the PhD Programs Foundation of Ministry of Education of China (no. 20093108110001), the Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (no. 09YZ13), Natural Science Foundation of Shanghai (11ZR1412500), Singapore MoE Tier 1 Research Grant no. RG60/07 and Shanghai Leading Academic Discipline Project (no. J50101). Israr Ali Khan’s research was also supported by the Higher Education Commission of Pakistan.