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 yuyaoming@online.sh.cn	College of Mathematics Science Shanghai Normal University Shanghai 200234 People's Republic of China grwang@shnu.edu.cn

Received 6 July 2006; revised 3 January 2007
Communicated by J. Koliha
Supported by Science Foundation of Shanghai Municipal Education Commission (CW0519).

Abstract

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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^†indicates author for correspondence

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