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.; Benítez López, J.; Wang, D. (2019). Centralizer's applications to the (b, c)-inverses in rings. Revista de la Real Academia de Ciencias Exactas, Físicas 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), 681–697 (2010)Benítez, J., Boasso, E.: The inverse along an element in rings with an involution, Banach algebras and $$C^*$$ C ∗ -algebras. Linear Multilinear Algebra 65(2), 284–299 (2017)Benítez, J., Boasso, E., Jin, H.W.: On one-sided $$(B, C)$$ ( B , C ) -inverses of arbitrary matrices. Electron. J. Linear Algebra 32, 391–422 (2017)Boasso, E., Kantún-Montiel, G.: The $$(b, c)$$ ( 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), 310–335 (2016)Chen, J.L., Ke, Y.Y., Mosić, D.: The reverse order law of the $$(b, c)$$ ( b , c ) -inverse in semigroups. Acta Math. Hung. 151(1), 181–198 (2017)Deng, C.Y.: Reverse order law for the group inverses. J. Math. Anal. Appl. 382(2), 663–671 (2011)Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)Drazin, M.P.: Left and right generalized inverses. Linear Algebra Appl. 510, 64–78 (2016)Jin, H.W., Benítez, J.: The absorption laws for the generalized inverses in rings. Electron. J. Linear Algebra 30, 827–842 (2015)Johnson, B.E.: An introduction to the theory of centralizers. Proc. Lond. Math. Soc. 14, 299–320 (1964)Ke, Y.Y., Cvetković-Ilić, D.S., Chen, J.L., Višnjić, J.: New results on $$(b, c)$$ ( b , c ) -inverses. Linear Multilinear Algebra 66(3), 447–458 (2018)Ke Y.Y., Višnjić J., Chen J.L.: One sided $$(b,c)$$ ( b , c ) -inverse in rings (2016). arXiv:1607.06230v1Liu, X.J., Jin, H.W., Cvetković-Ilić, D.S.: The absorption laws for the generalized inverses. Appl. Math. Comput. 219, 2053–2059 (2012)Mary, X.: On generalized inverse and Green’s relations. Linear Algebra Appl. 434, 1836–1844 (2011)Mary, X., Patrício, P.: Generalized inverses modulo $$\cal{H}$$ H in semigroups and rings. Linear Multilinear Algebra 61(8), 1130–1135 (2013)Mosić, D., Cvetković-Ilić, D.S.: Reverse order law for the Moore-Penrose inverse in $$C^*$$ C ∗ -algebras. Electron. J. Linear Algebra 22, 92–111 (2011)Rakić, D.S.: A note on Rao and Mitra’s constrained inverse and Drazin’s $$(b, c)$$ ( b , c ) -inverse. Linear Algebra Appl. 523, 102–108 (2017)Rakić, D.S., Dinčić, N.Č., Djordjević, D.S.: Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)Wang, L., Castro-González, N., Chen, J.L.: Characterizations of outer generalized inverses. Can. Math. Bull. 60(4), 861–871 (2017)Wei, Y.M.: A characterization and representation of the generalized inverse $$A^{(2)}_{T, S}$$ A T , S ( 2 ) and its applications. Linear Algebra Appl. 280, 87–96 (1998)Xu, S.Z., Benítez, J.: Existence criteria and expressions of the $$(b, c)$$ ( b , c ) -inverse in rings and its applications. Mediterr. J. Math. 15, 14 (2018)Zhu, H.H., Chen, J.L., Patrício, P.: Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64(3), 393–403 (2016)Zhu, H.H., Chen, J.L., Patrício, P.: Reverse order law for the inverse along an element. Linear Multilinear Algebra 65, 166–177 (2017)Zhu, H.H., Chen, J.L., Patrício, P., Mary, X.: Centralizer’s applications to the inverse along an element. Appl. Math. Comput. 315, 27–33 (2017)Zhu, H.H., Zhang, X.X., Chen, J.L.: Centralizers and their applications to generalized inverses. Linear Algebra Appl. 458, 291–300 (2014

Characterization and Representation of Weighted Core Inverse of Matrices

In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Several properties of these inverses and their interconnections with other generalized inverses are explored. Through one-sided core and dual-core inverse, the existence of a generalized weighted Moore-Penrose inverse of matrices is proposed. Further, by applying a new representation and using the properties of the weighted core inverse of a matrix, we discuss a few new results related to the reverse order law for these inverses.Comment: 18 page