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Linear and Multilinear Algebra Volume 59, 2011 - Issue 9
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Original Articles

The Drazin inverse in an arbitrary semiring

Israr Ali Khan Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
&
Qing-Wen Wang Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China; Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371Correspondence[email protected]
Pages 1019-1029 | Received 03 Aug 2010, Accepted 27 Dec 2010, Published online: 15 Apr 2011
 
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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.

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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.

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