A Note on Skew Generalized Power Serieswise Reversible Property

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-21-2023-69
Published: Jul 17, 2023
Cite as:
Eltiyeb Ali, A Note on Skew Generalized Power Serieswise Reversible Property, Int. J. Anal. Appl., 21 (2023), 69.

Main Article Content

Eltiyeb Ali

Abstract

The aim of this paper is to introduce and study (S, ω)-nil-reversible rings wherein we call a ring R is (S, ω)-nil-reversible if the left and right annihilators of every nilpotent element of R are equal. The researcher obtains various necessary or sufficient conditions for (S, ω)-nil-reversible rings are abelian, 2-primal, (S, ω)-nil-semicommutative and (S, ω)-nil-Armendariz. Also, he proved that, if R is completely (S, ω)-compatible (S, ω)-nil-reversible and J an ideal consisting of nilpotent elements of bounded index ≤ n in R, then R/J is (S, ¯ω)-nil-reversible. Moreover, other standard rings-theoretic properties are given.

Article Details

References

  1. P.M. Cohn, Reversible Rings, Bull. London Math. Soc. 31 (1999), 641-648. https://doi.org/10.1112/s0024609399006116.
  2. M.B. Rege, S. Chhawchharia, Armendariz Rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14-17. https://doi.org/10.3792/pjaa.73.14.
  3. J. Lambek, On the Representation of Modules by Sheaves of Factor Modules, Can. Math. Bull. 14 (1971), 359-368. https://doi.org/10.4153/cmb-1971-065-1.
  4. N.K. Kim, Y. Lee, Extensions of Reversible Rings, J. Pure Appl. Algebra. 185 (2003), 207-223. https://doi.org/10.1016/s0022-4049(03)00109-9.
  5. G. Yang, Z.K. Liu, On Strongly Reversible Rings, Taiwan. J. Math. 12 (2008), 129-136. https://www.jstor.org/stable/43833897.
  6. G. Marks, R. Mazurek, M. Ziembowski, A Unified Approach to Various Generalizations of Armendariz Rings, Bull. Aust. Math. Soc. 81 (2010), 361-397. https://doi.org/10.1017/s0004972709001178.
  7. P. Ribenboim, Noetherian Rings of Generalized Power Series, J. Pure Appl. Algebra. 79 (1992), 293-312. https://doi.org/10.1016/0022-4049(92)90056-l.
  8. R. Mazurek, M. Ziembowski, On Von Neumann Regular Rings of Skew Generalized Power Series, Commun. Algebra. 36 (2008), 1855-1868. https://doi.org/10.1080/00927870801941150.
  9. E. Ali, A. Elshokry, Some Results on a Generalization of Armendariz Rings, Asia Pac. J. Math. 6 (2019), 1. https://doi.org/10.28924/APJM/6-1.
  10. R. Antoine, Nilpotent Elements and Armendariz Rings, J. Algebra. 319 (2008), 3128-3140. https://doi.org/10.1016/j.jalgebra.2008.01.019.
  11. E. Ali, A. Elshokry, Extended of Generalized Power Series Reversible Rings, Italian J. Pure Appl. Math. In Press.
  12. S. Subba, T. Subedi, Nil-Reversible Rings, (2021). https://doi.org/10.48550/ARXIV.2102.11512.
  13. L. Ouyang, Extensions of Nilpotent P.P. Rings, Bull. Iran. Math. Soc. 36 (2010), 169-184.
  14. L. Ouyang, Special Weak Properties of Generalized Power Series Rings, J. Korean Math. Soc. 49 (2012), 687-701. https://doi.org/10.4134/JKMS.2012.49.4.687.
  15. P.P. Nielsen, Semi-Commutativity and the Mccoy Condition, J. Algebra. 298 (2006), 134-141. https://doi.org/10.1016/j.jalgebra.2005.10.008.
  16. S. Yang, X. Song, Extensions of McCoy Rings Relative to a Monoid, J. Math. Res. Exposition. 28 (2008), 659-665. https://doi.org/10.3770/j.issn:1000-341X.2008.03.028.
  17. J. Lambek, On the Representation of Modules by Sheaves of Factor Modules, Can. Math. Bull. 14 (1971), 359-368. https://doi.org/10.4153/cmb-1971-065-1.