Revista de la
Unión Matemática Argentina
Bott–Duffin $(e, f)$ inverses in $e$-symmetric rings
Shibo Shi, Yuheng Liu, and Long Wang

Volume 69, no. 2 (2026), pp. 485–497    

Published online (final version): August 21, 2026

https://doi.org/10.33044/revuma.5405

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Abstract

We investigate several properties of Bott–Duffin $(e, f)$ inverses in $e$-symmetric rings. In particular, we present new characterizations of Bott–Duffin $(e, f)$ inverses in terms of units in such rings. We also study the EP property of an element $a$ in an $ab$-symmetric ring, where $b$ is a generalized inverse of $a$. Finally, we consider the Bott–Duffin decomposition associated with idempotents.

References

  1. R. Bott and R. J. Duffin, On the algebra of networks, Trans. Amer. Math. Soc. 74 (1953), 99–109.  DOI  MR  Zbl
  2. N. Castro-González, J. Chen, and L. Wang, Further results on generalized inverses in rings with involution, Electron. J. Linear Algebra 30 (2015), 118–134.  DOI  MR  Zbl
  3. M. P. Drazin, A class of outer generalized inverses, Linear Algebra Appl. 436 no. 7 (2012), 1909–1923.  DOI  MR  Zbl
  4. G. Kafkas, B. Ungor, S. Halicioglu, and A. Harmanci, Generalized symmetric rings, Algebra Discrete Math. 12 no. 2 (2011), 72–84.  MR  Zbl
  5. Y. Ke and J. Chen, The Bott–Duffin $(e,f)$-inverses and their applications, Linear Algebra Appl. 489 (2016), 61–74.  DOI  MR  Zbl
  6. T. Y. Lam and P. P. Nielsen, Jacobson pairs and Bott–Duffin decompositions in rings, in Rings, modules and codes, Contemp. Math. 727, American Mathematical Society, Providence, RI, 2019, pp. 249–267.  DOI  MR  Zbl
  7. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359–368.  DOI  MR  Zbl
  8. G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra 174 no. 3 (2002), 311–318.  DOI  MR  Zbl
  9. F. Meng and J. Wei, $e$-symmetric rings, Commun. Contemp. Math. 20 no. 3 (2018), Article No. 1750039.  DOI  MR  Zbl
  10. F. Meng and J. Wei, Some properties of $e$-symmetric rings, Turkish J. Math. 42 no. 5 (2018), 2389–2399.  DOI  MR  Zbl
  11. F. Meng and J. Wei, $(g,e)$-symmetric rings, Algebra Colloq. 31 no. 2 (2024), 263–270.  DOI  MR  Zbl
  12. F. Meng, J. Wei, and R. Chen, Weak $e$-symmetric rings, Comm. Algebra 51 no. 7 (2023), 3042–3050.  DOI  MR  Zbl
  13. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278.  DOI  MR  Zbl
  14. W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra 27 no. 8 (1999), 3583–3592.  DOI  MR  Zbl
  15. L. Ouyang and H. Chen, On weak symmetric rings, Comm. Algebra 38 no. 2 (2010), 697–713.  DOI  MR  Zbl
  16. J. Wei, Generalized weakly symmetric rings, J. Pure Appl. Algebra 218 no. 9 (2014), 1594–1603.  DOI  MR  Zbl