Algebraic Basic Knowledge

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Algebraic Theory of Generalized Inverses
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Abstract

In this beginning chapter, we shall review some of the basic concepts and set up some notations for the subsequent chapters. The readers are assumed to be familiar with most of the basic knowledge on sets, groups, rings, fields and vector spaces.

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References

  1. V.P. Camillo, F.J. Costa-Cano, J.J. Simón, Relating properties of a ring and its ring of row and column finite matrices. J. Algebra 244(2), 435–449 (2001)

    Article  MathSciNet  Google Scholar 

  2. J.L. Chen, J. Cui, Two questions of L. Vǎs on ∗-clean rings. Bull. Austral. Math. Soc. 88, 499–505 (2013)

    Google Scholar 

  3. J.L. Chen, X.X. Zhang, Coherent Rings and FP-injective Rings (Science Press, Bei**g, 2014)

    Google Scholar 

  4. J. Cui, X.B. Yin, Some characterizations of ∗-regular rings. Comm. Algebra 45(2), 841–848 (2017)

    Article  MathSciNet  Google Scholar 

  5. J. Cui, X.B. Yin, A question on ∗-regular rings. Bull. Korean Math. Soc. 55(5), 1333–1338 (2018)

    MathSciNet  Google Scholar 

  6. K.R. Goodearl, Von Neumann Regular Rings. Monographs and Studies in Mathematics, 4 (Pitman, London, 1979)

    Google Scholar 

  7. R.Z. Han, J.L. Chen, Generalized inverses of matrices over rings. Chinese Q. J. Math. 7(4), 40–47 (1992)

    Google Scholar 

  8. R.E. Hartwig, K. Spindelböck, Matrices for which A and A commute. Linear Multilinear Algebra 14, 241–256 (1984)

    Article  MathSciNet  Google Scholar 

  9. N. Jacobson, Some remarks on one-sided inverses. Proc. Amer. Math. Soc. 1(3), 352–355 (1950)

    Article  MathSciNet  Google Scholar 

  10. Y.Y. Ke, J.L. Chen, P. Stanimirović, M. Ćirić, Characterizations and representations of outer inverse for matrices over a ring. Linear Multilinear Algebra 69(1), 155–176 (2021)

    Article  MathSciNet  Google Scholar 

  11. D. Khurana, T.Y. Lam, Commutators and anti-commutators of idempotents in rings. Contemp. Math. 715, 205–224 (2018)

    Article  MathSciNet  Google Scholar 

  12. J.J. Koliha, V. RakČević, Invertibility of the sum of idempotents. Linear Multilinear Algebra 50(4), 285–292 (2002)

    Article  MathSciNet  Google Scholar 

  13. J.J. Koliha, V. RakČević, Invertibility of the difference of idempotents. Linear Multilinear Algebra 51(1), 97–110 (2003)

    Article  MathSciNet  Google Scholar 

  14. J.J. Koliha, D.S. Cvetković-Ilić, C.Y. Deng, Generalized Drazin invertibility of combinations of idempotents. Linear Algebra Appl. 437, 2317–2324 (2012)

    Article  MathSciNet  Google Scholar 

  15. T.Y. Lam, Lectures on Modules and Rings (Springer, New York, 1999)

    Book  Google Scholar 

  16. T.Y. Lam, A First Course in Noncommutative Rings, 2nd edn. (Springer, New York, 2001)

    Book  Google Scholar 

  17. D.S. Rakić, N.Č. Dinčić, D.S. Djordjević, Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)

    Article  MathSciNet  Google Scholar 

  18. H.X. Wang, Core-EP decomposition and its applications. Linear Algebra Appl. 508, 289–300 (2016)

    Article  MathSciNet  Google Scholar 

  19. H.X. Wang, X.J. Liu, EP-nilpotent decomposition and its applications. Linear Multilinear Algebra 68(8), 1682–1694 (2020)

    Article  MathSciNet  Google Scholar 

  20. C. Wu, J.L. Chen, Left core inverses in rings with involution. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116(2), Page No. 67, 15 pp. (2022)

    Google Scholar 

  21. X.X. Zhang, J.L. Chen, Counterexamples in Ring Theory (Science Press, Bei**g, 2019)

    Google Scholar 

  22. X.X. Zhang, J.L. Chen, L. Wang, Generalized symmetric ∗-rings and Jacobson’s lemma for Moore-Penrose inverse. Publ. Math. Debrecen 91(3–4), 321–329 (2017)

    Article  MathSciNet  Google Scholar 

  23. H.H. Zhu, J.L. Chen, P. Patrício, Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64(3), 393–403 (2016)

    Article  MathSciNet  Google Scholar 

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Chen, J., Zhang, X. (2024). Algebraic Basic Knowledge. In: Algebraic Theory of Generalized Inverses. Springer, Singapore. https://doi.org/10.1007/978-981-99-8285-1_1

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