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Characterizations of Outer Generalized Inverses

Published online by Cambridge University Press:  20 November 2018

Long Wang ,
Nieves Castro-Gonzalez  and
Jianlong Chen
Long Wang
Affiliation:
Department of Mathematics, Taizhou University, Taizhou 225300, China e-mail: wanglseu@hotmail.com
Nieves Castro-Gonzalez
Affiliation:
Departamento Matemática Aplicada, ETSI Informática, Universidad Politécnica Madrid, 28660 Madrid, Spain e-mail: nieves@fi.upm.es
Jianlong Chen
Affiliation:
Department of Mathematics, Southeast University, Nanjing, 210096, China e-mail: wanglong19881208@sina.com
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Abstract

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Let $R$ be a ring and $b,c\in R$. In this paper, we give some characterizations of the $(b,c)$-inverse in terms of the direct sum decomposition, the annihilator, and the invertible elements. Moreover, elements with equal $(b,c)$-idempotents related to their $(b,c)$-inverses are characterized, and the reverse order rule for the $(b,c)$-inverse is considered.

Keywords

15A0916U99(b, c)-inverse(b, c)-idempotentregularityimage-kernel (p, q)-inversering
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 4 , 01 December 2017 , pp. 861 - 871
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Bott, R. and Duffin, R. J., On the algebra of networks. Trans. Amer. Math. Soc. 74(1953), 99–109. http://dx.doi.org/10.1090/S0002-9947-1953-0056573-X Google Scholar
[2] Cao, J. and Y. Xue, The characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras. Filomat 27(2013), 851–863. http://dx.doi.org/10.2298/FIL1305851C Google Scholar
[3] Cao, J., Perturbation analysis for the generalized inverses with prescribed idempotents in Banach algebras. arxiv:1301.4314Google Scholar
[4] Castro, N.-Gonzalez, J. Chen, and L. Wang, Further results on generalized inverses in rings with involution. Electron. J. Linear Algebra 30(2015), 118–134. http://dx.doi.org/1 0.13001 /1081 -381 0.2958 Google Scholar
[5] Castro, N.-Gonzalez, Koliha, J. J., and Y. Wei, Perturbation oftheDrazin inverse for matrices with equal eigenprojections at zero. Linear Algebra Appl. 312(2000), no. 1–3,181-189. http://dx.doi.org/1 0.101 6/SOO2 4-3795(00)00101 -4 Google Scholar
[6] Djordjevic, D. S. and Wei, Y. M., Outer generalized inverses in rings. Comm. Algebra 33(2005), no. 9, 3051-3060. http://dx.doi.org/10.1081/ACB-200066112 Google Scholar
[7] Drazin, M. P., A class of outer generalized inverses. Linear Algebra Appl. 436(2012), no. 7, 1909-1923. http://dx.doi.org/10.101 6/j.laa.2O11.09.004 Google Scholar
[8] Huylebrouck, D., R. Puystjens, and J. van Geel, The Moore-Penrose inverse of a matrix over a semi-simple artinian ring. Linear and Multilinear Algebra 16(1984), no. 1–4, 239–246. http://dx.doi.org/10.1080/03081088408817625 Google Scholar
[9] ILic, D.C., Liu, X. J., and J. Zhong, On the (p, q)-outer generalized inverse in Banach algebra. Appl. Math. Comput. 209(2009), 191–196. http://dx.doi.org/10.1016/j.amc.2008.12.031 Google Scholar
[10] Kantun-Montiel, G., Outer generalized inverses with prescribed ideals. Linear Multilinear Algebra 62(2014), no. 9, 1187–1196. http://dx.doi.org/10.1080/03081087.2013.816302 Google Scholar
[11] Koliha, J. J. and P. Patricio, Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72(2002), no. 1, 137–152. http://dx.doi.org/10.101 7/S1446788700003657 Google Scholar
[12] Mosic, D., Djordjevic, D. S., and G. Kantun-Montiel, Image-kernel (p, q)-inverse in rings. Electron. J. Linear Algebra 27(2014), 272–283. http://dx.doi.org/10.13001/1081 -3810.1 618 Google Scholar
[13] Patricio, P. and C. Mendes Araujo, Moore-Penrose invertibility in involutory rings: The case aaf = bbf. Linear Multilinear Algebra 58(2010), no. 3-4, 445-452. http://dx.doi.org/10.1080/03081080802652279 Google Scholar
[14] Patricio, P. and R. Puystjens, Generalized invertibility in two semigroups of a ring. Linear Algebra Appl. 377(2004), 125-139. http://dx.doi.org/10.1016/jJaa.2003.08.004 Google Scholar