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A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

  • Received : 2012.10.03
  • Published : 2014.01.01

Abstract

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

Keywords

References

  1. G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr, and F. Shaveisi, The classification of the annihilating-ideal graphs of commutative rings, Algebra Colloquium, (to appear).
  2. S. Akbari and A. Mohammadian, On zero-divisor graphs of finite rings, J. Algebra 314 (2007), no. 1, 168-184. https://doi.org/10.1016/j.jalgebra.2007.02.051
  3. D. F. Anderson, M. C. Axtell, and J. A. Stickles, Jr., Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson, Eds.), 23-45, Springer-Verlag, New York, 2011.
  4. D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, The zero-divisor graph of a commutative ring, II, in: Lecture Notes in Pure and Appl. Math., vol. 220, pp. 61-72, Dekker, New York, 2001.
  5. 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
  6. 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
  7. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
  8. M. Baziar, E. Momtahan, and S. Safaeeyan, A zero-divisor graph for modules with respect to their (first) dual, J. Algebra Appl. 12 (2013), no. 2, 1250151, 11 pp. https://doi.org/10.1142/S0219498812501514
  9. M. Baziar, E. Momtahan, and S. Safaeeyan, A zero-divisor graph for modules with respect to elements of their (first) dual, submitted to Bull. of IMS.
  10. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  11. M. Behboodi, Zero divisor graphs for modules over commutative rings, J. Commut. Algebra 4 (2012), no. 2, 175-197. https://doi.org/10.1216/JCA-2012-4-2-175
  12. M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727-739. https://doi.org/10.1142/S0219498811004896
  13. M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 740-753.
  14. D. Lu and T. Wu, On bipartite zero-divisor graphs, Discrete Math. 309 (2009), no. 4, 755-762. https://doi.org/10.1016/j.disc.2008.01.044
  15. J. Dauns and L. Fuchs, Infinite Goldie dimensions, J. Algebra 115 (1988), no. 2, 297-302. https://doi.org/10.1016/0021-8693(88)90257-8
  16. F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of commutative rings, Internat. J. Commutative Rings 1 (2002), no. 3, 93-106.
  17. S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533-3558. https://doi.org/10.1081/AGB-120004502
  18. S. P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings 1 (2002), no. 4, 203-211.
  19. D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River, 2001.
  20. R. Wisbauer, Foundations of Modules and Ring Theory, Gordon and Breach Reading 1991.

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  3. ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS vol.53, pp.5, 2016, https://doi.org/10.4134/JKMS.j150457