2 research outputs found
Centralizer's applications to the inverse along an element
In this paper, we firstly prove that the absorption law for one-sided inverses along an element holds, and derive the absorption law for the inverse along an element. We then obtain the absorption law for the inverse along different elements. Also, we prove that a left inverse of a along d coincides with a right inverse of a along d, provided that they both exist. Then, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the MooreâPenrose inverse of a regular element by one-sided invertibilities in a ring with involution.FCT - Natural Science Foundation of Jiangsu Province(UID-MAT-00013/2013)This research was carried out by the first author during his visit to the Department
of Mathematics and Applications, University of Minho, Portugal. He gratefully acknowledges
the financial support of China Scholarship Council. This research is also supported
11
by the National Natural Science Foundation of China (No. 11371089), the Specialized
Research Fund for the Doctoral Program of Higher Education (No. 20120092110020),
the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Scientific
Innovation Research of College Graduates in Jiangsu Province (No. CXLX13-072), the
Scientific Research Foundation of Graduate School of Southeast University, the FEDER
Funds through Programa Operacional Factores de Competitividade-COMPETEâ, the Portuguese
Funds through FCT- âFundažcËao para a CiËencia e a Tecnologiaâ, within the project
UID-MAT-00013/2013.info:eu-repo/semantics/publishedVersio
Centralizer's applications to the (b, c)-inverses in rings
[EN] We give several conditions in order that the absorption law for one sided (b,c)-inverses in rings holds. Also, by using centralizers, we obtain the absorption law for the (b,c)-inverse and the reverse order law of the (b,c)-inverse in rings. As applications, we obtain the related results for the inverse along an element, Moore-Penrose inverse, Drazin inverse, group inverse and core inverse.This research is supported by the National Natural Science Foundation of China (no. 11771076 and no. 11871301). The first author is grateful to China Scholarship Council for giving him a scholarship for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Chen, J.; BenĂtez LĂłpez, J.; Wang, D. (2019). Centralizer's applications to the (b, c)-inverses in rings. Revista de la Real Academia de Ciencias Exactas, FĂsicas y Naturales. 113(3):1739-1746. https://doi.org/10.1007/s13398-018-0574-0S173917461133Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681â697 (2010)BenĂtez, J., Boasso, E.: The inverse along an element in rings with an involution, Banach algebras and C â -algebras. Linear Multilinear Algebra 65(2), 284â299 (2017)BenĂtez, J., Boasso, E., Jin, H.W.: On one-sided ( B , C ) -inverses of arbitrary matrices. Electron. J. Linear Algebra 32, 391â422 (2017)Boasso, E., KantĂșn-Montiel, G.: The ( b , c ) -inverses in rings and in the Banach context. Mediterr. J. Math. 14, 112 (2017)Chen, Q.G., Wang, D.G.: A class of coquasitriangular Hopf group algebras. Comm. Algebra 44(1), 310â335 (2016)Chen, J.L., Ke, Y.Y., MosiÄ, D.: The reverse order law of the ( b , c ) -inverse in semigroups. Acta Math. Hung. 151(1), 181â198 (2017)Deng, C.Y.: Reverse order law for the group inverses. J. Math. Anal. Appl. 382(2), 663â671 (2011)Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506â514 (1958)Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909â1923 (2012)Drazin, M.P.: Left and right generalized inverses. Linear Algebra Appl. 510, 64â78 (2016)Jin, H.W., BenĂtez, J.: The absorption laws for the generalized inverses in rings. Electron. J. Linear Algebra 30, 827â842 (2015)Johnson, B.E.: An introduction to the theory of centralizers. Proc. Lond. Math. Soc. 14, 299â320 (1964)Ke, Y.Y., CvetkoviÄ-IliÄ, D.S., Chen, J.L., ViĆĄnjiÄ, J.: New results on ( b , c ) -inverses. Linear Multilinear Algebra 66(3), 447â458 (2018)Ke Y.Y., ViĆĄnjiÄ J., Chen J.L.: One sided ( b , c ) -inverse in rings (2016). arXiv:1607.06230v1Liu, X.J., Jin, H.W., CvetkoviÄ-IliÄ, D.S.: The absorption laws for the generalized inverses. Appl. Math. Comput. 219, 2053â2059 (2012)Mary, X.: On generalized inverse and Greenâs relations. Linear Algebra Appl. 434, 1836â1844 (2011)Mary, X., PatrĂcio, P.: Generalized inverses modulo H in semigroups and rings. Linear Multilinear Algebra 61(8), 1130â1135 (2013)MosiÄ, D., CvetkoviÄ-IliÄ, D.S.: Reverse order law for the Moore-Penrose inverse in C â -algebras. Electron. J. Linear Algebra 22, 92â111 (2011)RakiÄ, D.S.: A note on Rao and Mitraâs constrained inverse and Drazinâs ( b , c ) -inverse. Linear Algebra Appl. 523, 102â108 (2017)RakiÄ, D.S., DinÄiÄ, N.Ä., DjordjeviÄ, D.S.: Group, MooreâPenrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115â133 (2014)Wang, L., Castro-GonzĂĄlez, N., Chen, J.L.: Characterizations of outer generalized inverses. Can. Math. Bull. 60(4), 861â871 (2017)Wei, Y.M.: A characterization and representation of the generalized inverse A T , S ( 2 ) and its applications. Linear Algebra Appl. 280, 87â96 (1998)Xu, S.Z., BenĂtez, J.: Existence criteria and expressions of the ( b , c ) -inverse in rings and its applications. Mediterr. J. Math. 15, 14 (2018)Zhu, H.H., Chen, J.L., PatrĂcio, P.: Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64(3), 393â403 (2016)Zhu, H.H., Chen, J.L., PatrĂcio, P.: Reverse order law for the inverse along an element. Linear Multilinear Algebra 65, 166â177 (2017)Zhu, H.H., Chen, J.L., PatrĂcio, P., Mary, X.: Centralizerâs applications to the inverse along an element. Appl. Math. Comput. 315, 27â33 (2017)Zhu, H.H., Zhang, X.X., Chen, J.L.: Centralizers and their applications to generalized inverses. Linear Algebra Appl. 458, 291â300 (2014