Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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ISSN 2541-9005 (Online)

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Danchev P. V. Symmetrization in Clean and Nil-Clean Rings. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 2, pp. 154-160. DOI: 10.18500/1816-9791-2020-20-2-154-160, EDN: ZQRQJZ

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Symmetrization in Clean and Nil-Clean Rings

Danchev Peter Vasilevich, Bulgarian Academy of Sciences

We introduce and investigate D-clean and D-nil-clean rings as well as some other closely related symmetric versions of cleanness and nil-cleanness. A comprehensive structural characterization is given for these symmetrically clean and symmetrically nil-clean rings in terms of Jacobson radical and its quotient. It is proved that strongly clean (resp., strongly nil-clean) rings are always D-clean (resp., D-nil-clean).Our results corroborate our recent findings published in Bull. Irkutsk State Univ., Math. (2019) and Turk. J. Math. (2019). We also show that weakly nil-clean rings defined as in Danchev-McGovern (J. Algebra, 2015) and Breaz – Danchev – Zhou (J. Algebra and Appl., 2016) are actually weakly nil clean in the sense of Danchev-Ster (Taiwanese J. Math., 2015). This answers the question of the reviewer D. Khurana (Math. Review, 2017).

  1. Lam T. Y. A First Course in Noncommutative Rings. 2nd ed. (Graduate Texts in Math. Vol. 131). Berlin, Heidelberg, New York, Springer-Verlag, 2001. 388 p.
  2. Nicholson W. K. Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 1977, vol. 229, pp. 269–278.
  3. Diesl A. J. Nil clean rings. J. Algebra, 2013, vol. 383, pp. 197–211. DOI: https://doi.org/10.1016/j.jalgebra.2013.02.020
  4. Danchev P. V., McGovern W. Wm. Commutative weakly nil clean unital rings. J. Algebra, 2015, vol. 425, pp. 410–422. DOI: https://doi.org/10.1016/j.jalgebra.2014.12.003
  5. Danchev P. V. Left-right cleanness and nil cleanness in unital rings. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 27, pp. 28–35. DOI: https://doi.org/10.26516/1997-7670.2019.27.28
  6. Danchev P. V. A generalization of π-regular rings. Turk. J. Math., 2019, vol. 43, pp. 702– 711.
  7. Danchev P. V., Lam T. Y. Rings with unipotent units. Publ. Math. Debrecen, 2016, vol. 88, pp. 449–466. DOI: https://doi.org/10.5486/PMD.2016.7405
  8. Azumaya G. Strongly π-regular rings. J. Fac. Sci. Hokkaido Univ. (Ser. I, Math.), 1954, vol. 13, pp. 34–39.
  9. Nicholson W. K. Strongly clean rings and Fitting’s lemma. Commun. Algebra, 1999, vol. 27, pp. 3583–3592.
  10. Danchev P. V. Generalizing nil clean rings. Bull. Belg. Math. Soc. Simon Stevin, 2018, vol. 25, no. 1, pp. 13–29. DOI: https://doi.org/10.36045/bbms/1523412048
  11. Ster J. Nil-clean quadratic elements. J. Algebra and Appl., 2017, vol. 16, no. 10, p. 1750197. DOI: https://doi.org/10.1142/S0219498817501973
  12. Breaz S., Danchev P., Zhou Y. Rings in which every element is either a sum or a difference of a nilpotent and an idempotent. J. Algebra and Appl., 2016, vol. 15, no. 08, p. 1650148. DOI: https://doi.org/10.1142/S0219498816501486
  13. Danchev P., Ster J. Generalizing π-regular rings. Taiwanese J. Math., 2015, vol. 19, no. 6, pp. 1577–1592. DOI: https://doi.org/10.11650/tjm.19.2015.6236
  14. Khurana D. Math. Review 3528770 (2017).
  15. Danchev P. V. Weakly UU rings. Tsukuba J. Math., 2016, vol. 40, no. 1, pp. 101–118.
  16. Kosan M. T., Yildirim T., Zhou Y. Rings with x n − x nilpotent. J. Algebra and Appl., 2020, vol. 19. DOI: https://doi.org/10.1142/S0219498820500656
  17. Hirano Y., Tominaga H., Yaqub A. On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element. Math. J. Okayama Univ., 1988, vol. 30, pp. 33–40.