Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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WEAKLY (m, n)-CLOSED IDEALS AND (m, n)-VON NEUMANN REGULAR RINGS

  • Anderson, David F. (Department of Mathematics The University of Tennessee) ;
  • Badawi, Ayman (Department of Mathematics & Statistics The American University of Sharjah) ;
  • Fahid, Brahim (Department of Mathematics Faculty of Sciences, B.P. 1014 Mohammed V University)
  • Received : 2017.05.21
  • Accepted : 2018.06.26
  • Published : 2018.09.01
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Abstract

Let R be a commutative ring with $1{\neq}0$, I a proper ideal of R, and m and n positive integers. In this paper, we define I to be a weakly (m, n)-closed ideal if $0{\neq}x^m\;{\in}I$ for $x{\in}R$ implies $x^n{\in}I$, and R to be an (m, n)-von Neumann regular ring if for every $x{\in}R$, there is an $r{\in}R$ such that $x^mr=x^n$. A number of results concerning weakly(m, n)-closed ideals and (m, n)-von Neumann regular rings are given.

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References

  1. D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003), no. 4, 831-840.
  2. D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39 (2011), no. 5, 1646-1672. https://doi.org/10.1080/00927871003738998
  3. D. F. Anderson and A. Badawi, Von Neumann regular and related elements in commutative rings, Algebra Colloq. 19 (2012), Special Issue no. 1, 1017-1040. https://doi.org/10.1142/S1005386712000831
  4. D. F. Anderson and A. Badawi, On (m, n)-closed ideals of commutative rings, J. Algebra Appl. 16 (2017), no. 1, 1750013, 21 pp. https://doi.org/10.1142/S021949881750013X
  5. A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 417-429. https://doi.org/10.1017/S0004972700039344
  6. A. Badawi, On weakly semiprime ideals of commutative rings, Beitr. Algebra Geom. 57 (2016), no. 3, 589-597. https://doi.org/10.1007/s13366-016-0283-9
  7. A. Badawi, n-absorbing ideals of commutative rings and recent progress on three conjectures: a survey, in Rings, polynomials, and modules, 33-52, Springer, Cham, 2017.
  8. A. Badawi and B. Fahid, On weakly 2-absorbing ${\delta}$-primary ideals in commutative rings, to appear in Georgian Math. J..
  9. A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc. 51 (2014), no. 4, 1163-1173. https://doi.org/10.4134/BKMS.2014.51.4.1163
  10. A. Badawi, U. Tekir, and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc. 52 (2015), no. 1, 97-111. https://doi.org/10.4134/JKMS.2015.52.1.097
  11. A. Badawi and A. Yousefian Darani, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39 (2013), no. 2, 441-452.
  12. S. Ebrahimi Atani and F. Farzalipour, On weakly primary ideals, Georgian Math. J. 12 (2005), no. 3, 423-429.
  13. B. Fahid and D. Zhao, 2-absorbing ${\delta}$-primary ideals in commutative rings, Kyungpook Math. J. 57 (2017), no. 2, 193-198. https://doi.org/10.5666/KMJ.2017.57.2.193
  14. J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
  15. H. Mostafanasab, F. Soheilnia, and A. Y. Darani, On weakly n-absorbing ideals of commutative rings, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 62 (2016), no. 2, vol. 3, 845-862.