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ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS

  • Received : 2015.08.02
  • Published : 2016.09.01

Abstract

Let M be an R-module, where R is a commutative ring with identity 1 and let G(V,E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$. We show that M over R is finite if and only if the metric dimension of the graph $ann_f({\Gamma}(M_R))$ is finite. We further show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if M is a prime-multiplication-like R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if $$M{\sim_=}R$$. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M) is a prime ideal and Soc(M) = 0.

Acknowledgement

Supported by : UGC New Delhi

References

  1. S. Akbari and A. Mohammadian, On zero-divisor graph of finite rings, J. Algebra 314 (2007), no. 1, 168-184. https://doi.org/10.1016/j.jalgebra.2007.02.051
  2. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
  3. D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500-517. https://doi.org/10.1006/jabr.1993.1171
  4. 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
  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. R. F. Bailey, J. Caceres, D. Garijo, A. Gonzalez, A. Marquezc, K. Meagherd, and M. L. Puertas, Resolving sets for Johnson and Kneser graphs, European J. Combin. 34 (2013), no. 4, 736-751. https://doi.org/10.1016/j.ejc.2012.10.008
  7. R. F. Bailey and P. J. Cameron, Base size, metric dimension and other invariants of groups and graphs, Bull. Lond. Math. Soc. 43 (2011), no. 2, 209-242. https://doi.org/10.1112/blms/bdq096
  8. R. F. Bailey and K. Meagher, On the metric dimension of Grassmann graphs, Discrete Math. Theor. Comput. Sci. 13 (2011), no. 4, 97-104.
  9. 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
  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. F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of commutative rings, Internat. J. Commutative Rings 1 (2002), no. 3, 93-106.
  12. J. Diaz, O. Pottonen, M. Serna, and E. J. van Leeuwen, On the complexity of metric dimension, On the complexity of metric dimension. , 419-430, Lecture Notes in Comput. Sci., 7501, Springer, Heidelberg, 2012.
  13. R. Diestel, Graph Theory, Springer-Verlag, New York, 1997.
  14. P. Erdos and A. Renyi, On two problems of information theory, Magyar Tud. Akad. Mat. Kutato Int. Kozl. 8 (1963), 229-243.
  15. M. Feng and K. Wang, On the metric dimension of bilinear forms graphs, Discrete Math. 312 (2012), no. 6, 1266-1268. https://doi.org/10.1016/j.disc.2011.11.020
  16. Sh. Ghalandarzadeh and P. Malakooti Rad, Torsion graph over multiplication modules, Extracta Math. 24 (2009), no. 3, 281-299.
  17. A. Haghany and M. R. Vedadi, Endoprime modules, Acta Math. Hungar. 106 (2005), no. 1-2, 89-99. https://doi.org/10.1007/s10474-005-0008-2
  18. F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191-195.
  19. S. Hoffmann and E Wanke, Metric dimension for Gabriel unit disk graphs is NP-complete, Lecture Notes Comp. Sci. 7718 (2013), 90-92.
  20. S. Khuller, B. Raghavachari, and A. Rosenfeld, Landmarks in graphs. Discrete Appl. Math. 70 (1996), no. 3, 217-229. https://doi.org/10.1016/0166-218X(95)00106-2
  21. 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
  22. S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533-3558. https://doi.org/10.1081/AGB-120004502
  23. S. Pirzada, An Introduction to Graph Theory, Universities Press, OrientBlackSwan, Hyderabad, India, 2012.
  24. S. Pirzada and R. Raja, On the metric dimension of a zero-divisor graph, Comm. Algebra, to appear.
  25. S. Pirzada, R. Raja, and S. P. Redmond, Locating sets and numbers of graphs associated to commutative rings, J. Algebra Appl. 13 (2014), no. 7, 1450047, 18 pages.
  26. R. Raja, S. Pirzada, and S. P. Redmond, On locating numbers and codes of zero-divisor graphs associated with commutative rings, J. Algebra Appl. 15 (2016), no. 1, 1650014, 22 pages.
  27. S. P. Redmond, The zero-divisor graph of a noncommutative ring, Internat. J. Commutative Rings 1 (2002), no. 4, 203-211.
  28. S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31 (2003), no. 9, 4425-4443. https://doi.org/10.1081/AGB-120022801
  29. S. Safaeeyan, M. Bazair and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc. 51 (2014), no. 1, 87-98. https://doi.org/10.4134/JKMS.2014.51.1.087
  30. D. B. West, Introduction to Graph Theory, 2nd ed. USA: Prentice Hall, 2001.
  31. R. Wisbauer, Foundations of Modules and Ring Theory, Gordon and Breach Reading, 1991.