Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EXTENDED ZERO-DIVISOR GRAPHS OF IDEALIZATIONS

  • Bennis, Driss (Laboratory of Analysis, Algebra and Decision Support Department of Mathematics Faculty of Sciences of Rabat, Mohammed V University in Rabat) ;
  • Mikram, Jilali (Laboratory of Mathematics, Computing and Applications, Department of Mathematics Faculty of Sciences of Rabat, Mohammed V University in Rabat) ;
  • Taraza, Fouad (Laboratory of Mathematics, Computing and Applications, Department of Mathematics Faculty of Sciences of Rabat, Mohammed V University in Rabat)
  • Received : 2016.03.10
  • Published : 2017.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a commutative ring with zero-divisors Z(R). The extended zero-divisor graph of R, denoted by $\bar{\Gamma}(R)$, is the (simple) graph with vertices $Z(R)^*=Z(R){\backslash}\{0\}$, the set of nonzero zero-divisors of R, where two distinct nonzero zero-divisors x and y are adjacent whenever there exist two non-negative integers n and m such that $x^ny^m=0$ with $x^n{\neq}0$ and $y^m{\neq}0$. In this paper, we consider the extended zero-divisor graphs of idealizations $R{\ltimes}M$ (where M is an R-module). At first, we distinguish when $\bar{\Gamma}(R{\ltimes}M)$ and the classical zero-divisor graph ${\Gamma}(R{\ltimes}M)$ coincide. Various examples in this context are given. Among other things, the diameter and the girth of $\bar{\Gamma}(R{\ltimes}M)$ are also studied.

Keywords

References

  1. D. F. Anderson, On the diameter and girth of a zero-divisor graph. II, Houston J. Math. 34 (2008), no. 2, 361-371.
  2. D. F. Anderson, M. Axtell, and J. Stickles, Zero-divisor graphs in commutative rings, In: M. Fontana, S.-E. Kabbaj, B. Olberding and I. Swanson, eds. Commutative Algebra, Noetherian and Non-Noetherian Perspectives, pp. 23-45, New York, Springer-Verlag, 2010.
  3. D. F. Anderson and A. Badawi, Divisibility conditions in commutative rings with zero-divisors, Comm. Algebra 30 (2002), no. 8, 4031-4047. https://doi.org/10.1081/AGB-120005834
  4. D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra 36 (2008), no. 8, 3073-3092. https://doi.org/10.1080/00927870802110888
  5. D. F. Anderson, R. Levy, and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221-241. https://doi.org/10.1016/S0022-4049(02)00250-5
  6. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434-447. https://doi.org/10.1006/jabr.1998.7840
  7. D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007), no. 2, 543-550. https://doi.org/10.1016/j.jpaa.2006.10.007
  8. D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500-514. https://doi.org/10.1006/jabr.1993.1171
  9. D. D. Anderson and M.Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
  10. M. Axtell, J. Coykendall, and J. Stickles, Zero-divisor graphs of polynomial and power series over commutative rings, Comm. Algebra 33 (2005), no. 6, 2043-2050. https://doi.org/10.1081/AGB-200063357
  11. M. Axtell and J. Stickles, Zero-divisor graphs of idealizations, J. Pure Appl. Algebra 204 (2006), no. 2, 235-243. https://doi.org/10.1016/j.jpaa.2005.04.004
  12. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  13. D. Bennis, J. Mikram, and F. Taraza, On the extended zero-divisor graph of commutative rings, Turk. J. Math. 40 (2016), no. 2, 376-388. https://doi.org/10.3906/mat-1504-61
  14. B. Bollabos, Modern Graph Theory, New York, Springer-Verlag, 1998.
  15. J. A. Huckaba, Commutative Rings with Zero Divisors, New York/Basel, Marcel Dekker, 1988.
  16. R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra 30 (2002), no. 2, 745-750. https://doi.org/10.1081/AGB-120013178
  17. T. G. Lucas, The diameter of a zero divisor graph, J. Algebra 301 (2006), no. 1, 174-193. https://doi.org/10.1016/j.jalgebra.2006.01.019