Table of Contents
ISRN Algebra
Volume 2012, Article ID 282054, 13 pages
http://dx.doi.org/10.5402/2012/282054
Research Article

When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?

Department of Mathematics, Saurashtra University, Rajkot 360 005, India

Received 12 March 2012; Accepted 3 April 2012

Academic Editors: D. Anderson, A. V. Kelarev, and C. Munuera

Copyright © 2012 S. Visweswaran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. F. Anderson and P. S. Livingston, β€œThe zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 217, no. 2, pp. 434–447, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. D. F. Anderson, M. C. Axtell, and J. A. Stickles, β€œZero-divisor graphs in commutative rings,” in Commutative Algebra: Noetherian and Non-Noetherian Perspectives, pp. 23–45, Springer, New York, NY, USA, 2011. View at Google Scholar
  3. D. F. Anderson, R. Levy, and J. Shapiro, β€œZero-divisor graphs, von Neumann regular rings, and Boolean algebras,” Journal of Pure and Applied Algebra, vol. 180, no. 3, pp. 221–241, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. R. Levy and J. Shapiro, β€œThe zero-divisor graph of von Neumann regular rings,” Communications in Algebra, vol. 30, no. 2, pp. 745–750, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New York, NY, USA, 2000.
  6. S. Visweswaran, β€œSome results on the complement of the zero-divisor graph of a commutative ring,” Journal of Algebra and its Applications, vol. 10, no. 3, pp. 573–595, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. S. Visweswaran, β€œSome properties of the complement of zero-divisor graph of a commutative ring,” ISRN Algebra, vol. 2011, Article ID 591041, 24 pages, 2011. View at Publisher Β· View at Google Scholar
  8. W. Heinzer and J. Ohm, β€œOn the Noetherian-like rings of E. G. Evans,” Proceedings of the American Mathematical Society, vol. 34, no. 1, pp. 73–74, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, Ill, USA, 1974.
  10. W. Heinzer and J. Ohm, β€œLocally noetherian commutative rings,” Transactions of the American Mathematical Society, vol. 158, pp. 273–284, 1971. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, β€œThe zero-divisor graph of a commutative ring. II,” in Ideal Theoretic Methods in Commutative Algebra, vol. 220 of Lecture Notes in Pure and Applied Mathematics,, pp. 61–72, Marcel Dekker, New York, NY, USA, 2001. View at Google Scholar Β· View at Zentralblatt MATH
  12. I. Beck, β€œColoring of commutative rings,” Journal of Algebra, vol. 116, no. 1, pp. 208–226, 1988. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
  14. N. Ganesan, β€œProperties of rings with a finite number of zero divisors,” Mathematische Annalen, vol. 157, pp. 215–218, 1964. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. A. V. Kelarev, Ring Constructions and Applications, vol. 9 of Series in Algebra, World Scientific, River Edge, NJ, USA, 2002.
  16. A. Kelarev, Graph Algebras and Automata, vol. 257, Marcel Dekker, New York, NY, USA, 2003.