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Equivalence Relations Based on (b,c)-Inverses in Rings

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Abstract

In this paper, we introduce a binary relation in the set of all (b, c)-invertible elements of a ring, defined in the manner like minus, star, sharp, core, and dual core partial orders. We prove that this binary relation is actually an equivalence relation and we investigate some of its properties. Furthermore, we define another equivalence relation on the set of all (b, c)-invertible elements of a ring, as an improvement of the previous mentioned equivalence relation. In addition, the equivalence criteria are given for the equality of (b, c)-related idempotents of two elements and their (b, c)-invertibility is studied.

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References

  1. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)

    Google Scholar 

  2. Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2012)

    Article  MathSciNet  Google Scholar 

  3. Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)

    Article  MathSciNet  Google Scholar 

  4. Drazin, M.P.: Natural structures on semigroups with involution. Bull. Am. Math. Soc. 84, 139–141 (1978)

    Article  MathSciNet  Google Scholar 

  5. Hartwig, R.E.: How to partially order regular elements. Math. Jpn. 25, 1–13 (1980)

    MathSciNet  Google Scholar 

  6. Ke,Y. Cvetković Ilić D. S. ,Chen J., Višnjić J. , New results on (b,c) - inverses, Linear Multilinear Algebra 66 (3) (2018), 447–458

  7. Ke, Y., Višnjić, J., Chen, J.: One-sided (b, c)-inverses in rings. Filomat 34(3), 727–736 (2020)

    Article  MathSciNet  Google Scholar 

  8. Mitra, S.K.: On group inverses and the sharp order. Linear Algebra Appl. 92, 17–37 (1987)

    Article  MathSciNet  Google Scholar 

  9. Mitsch, H.: A natural partial order for semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)

    Article  MathSciNet  Google Scholar 

  10. Mosić, D., Zou, H., Chen, J.: On the (b, c)-inverses in rings. Filomat 32(4), 1221–1231 (2018)

    Article  MathSciNet  Google Scholar 

  11. Nambooripad, K.S.S.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)

    Article  MathSciNet  Google Scholar 

  12. Rakić, D.S., Djordjević, D.S.: Star, sharp, core and dual core partial order in rings with involution. Appl. Math. Compt. 259, 800–818 (2015)

    Article  MathSciNet  Google Scholar 

  13. Rakić, D.S.: A note on Rao and Mitra’s constrained inverse and Drazin’s (b, c) inverse. Linear Algebra Appl. 523, 102–108 (2017)

    Article  MathSciNet  Google Scholar 

  14. Rao C. R. ,Mitra S. K. , Generalized inverse of a matrix and its application, in: Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., vol 1 (1972), 601–620

  15. Xu, S., Benítez, J.: Existence criteria and expressions of the (b, c)-inverse in rings and their applications. Mediterr. J. Math. 15, 14 (2018)

    Article  MathSciNet  Google Scholar 

  16. Xu, S., Chen, J., Benítez, J.: Projections for generalized inverses. Linear Multilinear Algebra 66(8), 1593–1605 (2018)

    Article  MathSciNet  Google Scholar 

  17. Wang, L., Castro-González, N., Chen, J.: Characterizations of outer generalized inverses. Canad. Math. Bull. 60, 861–871 (2017)

    Article  MathSciNet  CAS  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referees for their useful comments which greatly improved the presentation of this paper.

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Authors and Affiliations

  1. Technical Faculty in Bor, University of Belgrade, Vojske Jugoslavije 12, 19210, Bor, Serbia

    Ivana Stanišev

  2. Faculty of Medicine, University of Niš, Bulevar dr Zorana Djindjića 81, 18108, Niš, Serbia

    Jelena Višnjić

  3. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, P.O. Box 95, 18000, Niš, Serbia

    Dragan S. Djordjević

Authors
  1. Ivana Stanišev
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  2. Jelena Višnjić
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  3. Dragan S. Djordjević
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Contributions

Ivana Stanišev and Jelena Višnjić wrote the main manuscript text. Dragan Djordjević supervised and reviewed the work.

Corresponding author

Correspondence to Dragan S. Djordjević.

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I. Stanišev: Supported by Grant No. 451-03-47/2023-01/200131 of the Ministry of Education, Science and Technological Development, Republic of Serbia

J. Višnjić: Supported by Grant No. 451-03-47/2023-01/200113 of the Ministry of Education, Science and Technological Development, Republic of Serbia

D. S. Djordjević: Supported by Grant No. 451-03-47/2023–01/200124 of the Ministry of Education, Science and Technological Development, Republic of Serbia.

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Stanišev, I., Višnjić, J. & Djordjević, D.S. Equivalence Relations Based on (b,c)-Inverses in Rings. Mediterr. J. Math. 21, 11 (2024). https://doi.org/10.1007/s00009-023-02545-5

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  • DOI: https://doi.org/10.1007/s00009-023-02545-5

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