J. Aust. Math. Soc. 83 (2007), no. 3, pp. 423–437.

The generalized inverse A_{T,S}^{(2)} of a matrix over an associative ring

Yaoming Yu Guorong Wang
College of Education Shanghai Normal University Shanghai 200234 People's Republic of China
College of Mathematics Science Shanghai Normal University Shanghai 200234 People's Republic of China
Received 6 July 2006; revised 3 January 2007
Communicated by J. Koliha
Supported by Science Foundation of Shanghai Municipal Education Commission (CW0519).


In this paper we establish the definition of the generalized inverse A_{T,S}^{(2)} which is a \{2\} inverse of a matrix A with prescribed image T and kernel S over an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverse A_{T,S}^{(1,2)} and some explicit expressions for A_{T,S}^{(1,2)} of a matrix A over an associative ring, which reduce to the group inverse or \{1\} inverses. In addition, we show that for an arbitrary matrix A over an associative ring, the Drazin inverse A_{d}, the group inverse A_{g} and the Moore–Penrose inverse A^{\dagger }, if they exist, are all the generalized inverse A_{T,S}^{(2)}.

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2000 Mathematics Subject Classification: primary 15A33, 15A09
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