Abstract
In 1978, Campbell and Meyer proposed the notion of minimal rank weak Drazin inverses of complex matrices. In this paper, we define minimal weak Drazin inverses of elements in semigroups using Green’s preorder \(\leqslant _{{\mathcal {R}}},\) which generalize minimal rank weak Drazin inverses of complex matrices. For two elements a, y of a semigroup, it is proved that y is a minimal weak Drazin inverse of a if and only if \(ya^{k+1}=a^{k}\) for some nonnegative integer k and \(ay^{2}=y.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicable.
References
Azumaya, G.: Strongly \(\pi \)-regular rings. J. Fac. Sci. Hokkaido Univ. Ser. I(13), 34–39 (1954)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)
Campbell, S.L.: Singular linear systems of differential equations with delays. Appl. Anal. 11(2), 129–136 (1980)
Campbell, S.L., Meyer, C.D.: Weak Drazin inverses. Linear Algebra Appl. 20(2), 167–178 (1978)
Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. SIAM, Philadelphia (2009)
Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)
Drazin, M.P.: Generalizations of Fitting’s lemma in arbitrary associative rings. Commun. Algebra 29(8), 3647–3675 (2001)
Drazin, M.P.: Hybrid \((b, c)\)-inverses and five finiteness properties in rings, semigroups. and categories. Commun. Algebra 49(5), 2265–2277 (2021)
Gao, Y.F., Chen, J.L.: Pseudo core inverses in rings with involution. Commun. Algebra 64(1), 38–50 (2018)
Goyal, D.R., Khurana, A., Khurana, D.: Strongly \(\pi \)-regular elements and Drazin inverses. J. Algebra Appl. (2023). https://doi.org/10.1142/S0219498825500434
Green, J.A.: On the structure of semigroups. Ann. Math. 54, 163–172 (1951)
Jacobson, N.: The radical and semi-simplicity for arbitrary rings. Am. J. Math. 67, 300–320 (1945)
Lam, T.Y.: An introduction to \(q\)-central idempotents and \(q\)-abelian rings. Commun. Algebra 51(3), 1071–1088 (2023)
Mary, X.: On generalized inverses and Green’s relations. Linear Algebra Appl. 434(8), 1836–1844 (2011)
Nicholson, W.K.: Strongly clean rings and Fitting’s lemma. Commun. Algebra 27(8), 3583–3592 (1999)
Srivastava, S., Stanimirović, P.S., Katsikis, V.N., Gupta, D.K.: A family of iterative methods with accelerated convergence for restricted linear system of equations. Mediterr. J. Math. 14(6), 222, 26 (2017)
Wu, C., Chen, J.L.: Minimal rank weak Drazin inverses: a class of outer inverses with prescribed range. Electron. J. Linear Algebra 39, 1–16 (2023)
Zhou, Y.K., Chen, J.L.: Weak core inverses and pseudo core inverses in a ring with involution. Linear Multilinear Algebra 70(21), 6876–6890 (2022)
Acknowledgements
The authors wish to thank the editor and reviewers sincerely for their constructive comments and suggestions that have improved the quality of the paper. This research was supported by the National Natural Science Foundation of China (No. 12171083) and the Qing Lan Project of Jiangsu Province.
Author information
Authors and Affiliations
Contributions
S.X. Xu conceived the initial idea for the manuscript, and subsequently wrote and revised the entire manuscript. As the supervisor of S.X. Xu and corresponding author of this manuscript, J.L. Chen provided much guidance on the entire manuscript. C. Wu suggested revisions to the second part and read the entire manuscript. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xu, S., Chen, J. & Wu, C. Minimal Weak Drazin Inverses in Semigroups and Rings. Mediterr. J. Math. 21, 119 (2024). https://doi.org/10.1007/s00009-024-02661-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-024-02661-w