Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

An Alternative Perspective of Near-rings of Polynomials and Power series

  • Shokuhifar, Fatemeh (Faculty of Mathematical Sciences, Shahrood University of Technology) ;
  • Hashemi, Ebrahim (Faculty of Mathematical Sciences, Shahrood University of Technology) ;
  • Alhevaz, Abdollah (Faculty of Mathematical Sciences, Shahrood University of Technology)
  • Received : 2021.06.26
  • Accepted : 2022.01.13
  • Published : 2022.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if annS(a) = annS(b), is an equivalence relation. The compressed zero-divisor graph 𝚪E(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]S and [1]S, in which two distinct vertices [a]S and [b]S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R0[x] and the near-ring of the power series R0[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(𝚪E(R0[x])) and diam(𝚪E(R0[[x]])). As a corollary, it is shown that for a reduced ring R, diam(𝚪E(R)) ≤ diam(𝚪E(R0[x])) ≤ diam(𝚪E(R0[[x]])).

Keywords

References

  1. A. Alhevaz, E. Hashemi and F. Shokuhifar, On zero-divisor of near-rings of polynomials, Quaest. Math., 42(3)(2019), 363-372. https://doi.org/10.2989/16073606.2018.1455070
  2. D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero-divisor graph, J. Algebra, 447(2016), 297-321. https://doi.org/10.1016/j.jalgebra.2015.08.021
  3. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(1999), 434-447. https://doi.org/10.1006/jabr.1998.7840
  4. E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc., 18(1974), 470-473. https://doi.org/10.1017/S1446788700029190
  5. I. Beck, Coloring of commutative rings, J. Algebra, 116(1988), 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  6. G. F. Birkenmeier and F. K. Huang, Annihilator conditions on formal power series, Algebra Colloq., 9(1)(2002), 29-37.
  7. G. F. Birkenmeier and F. K. Huang, Annihilator conditions on polynomials, Comm. Algebra, 29(5)(2001), 2097-2112. https://doi.org/10.1081/AGB-100002172
  8. G. A. Cannon, K. M. Neuerburg, and S. P. Redmond, Zero-divisor graphs of nearrings and semigroups, Nearrings and nearfields, Springer, Dordrecht(2005).
  9. C. Faith, Annihilators, associated prime ideals and Kasch-McCoy commutative rings, Comm. Algebra, 119(1991), 1867-1892.
  10. D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc., 27(1971), 427-433. https://doi.org/10.1090/S0002-9939-1971-0271100-6
  11. E. Hashemi, On nilpotent elements in a near-ring of polynomials, Math. Commun., 17(2012), 257-264.
  12. E. Hashemi, M. Abdi and A. Alhevaz, On the diameter of the compressed zero-divisor graph, Comm. Algebra, 45(2017), 4855-4864. https://doi.org/10.1080/00927872.2017.1284227
  13. E. Hashemi and A. Moussavi, Skew power series extensions of α-rigid p.p. rings, Bull. Korean math. Soc., 41(4)(2004), 657-664. https://doi.org/10.4134/BKMS.2004.41.4.657
  14. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168(1)(2002), 45-52. https://doi.org/10.1016/S0022-4049(01)00053-6
  15. J. A. Huckaba, Commutative Rings with Zero-Divisors, Marcel Dekker Inc., New York(1988).
  16. Z. Liu and R. Zhao, On weak Armendariz rings, Comm. Algebra, 34(2006), 2607-2616. https://doi.org/10.1080/00927870600651398
  17. T. Lucas, The diameter of a zero-divisor graph, J. Algebra, 301(2006), 174-193. https://doi.org/10.1016/j.jalgebra.2006.01.019
  18. H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34(2006), 923-929. https://doi.org/10.1080/00927870500441858
  19. N. H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly, 49(1942), 286-295. https://doi.org/10.2307/2303094
  20. S. B. Mulay, Cycles and symmetries of zero-divisor, Comm. Algebra, 30(2002), 3533-3558. https://doi.org/10.1081/AGB-120004502
  21. G. Pilz, Near-rings, second edition, North-Holland Mathematics Studies, 23, North-Holland Publishing Co., Amsterdam(1983).
  22. S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings, 1(2002), 203-211.
  23. S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(2011), 2338-2348. https://doi.org/10.1080/00927872.2010.488675