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ZERO-DIVISOR GRAPHS WITH RESPECT TO PRIMAL AND WEAKLY PRIMAL IDEALS

  • Published : 2009.03.31

Abstract

We consider zero-divisor graphs with respect to primal, nonprimal, weakly prime and weakly primal ideals of a commutative ring R with non-zero identity. We investigate the interplay between the ringtheoretic properties of R and the graph-theoretic properties of ${\Gamma}_I(R)$ for some ideal I of R. Also we show that the zero-divisor graph with respect to primal ideals commutes by localization.

Keywords

References

  1. D. F. Anderson, R. Levy, and J. Shapirob, 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
  2. 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
  3. D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003), no. 4, 831–840.
  4. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
  5. L. Dancheng and W. Tongsuo, On ideal-based zero-divisor graphs, Preprint 2006.
  6. S. Ebrahimi Atani and A. Yousefian Darani, Some remarks on primal submodules, Sarajevo Journal of Mathematics 4 (2008), no. 17, 181–190.
  7. S. Ebrahimi Atani and A. Yousefian Darani, On weakly primal ideals (I), Demonstratio Math. 40 (2007), no. 1, 23–32.
  8. L. Fuchs, On primal ideals, Proc. Amer. Math. Soc. 1 (1950), 1–6. https://doi.org/10.2307/2032421
  9. J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117. Marcel Dekker, Inc., New York, 1988.
  10. 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
  11. H. R. Maimani, M. R. Pournaki, and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra 34 (2006), no. 3, 923–929. https://doi.org/10.1080/00927870500441858
  12. S. B. Mulay, Rings having zero-divisor graphs of small diameter or large girth, Bull. Austral. Math. Soc. 72 (2005), no. 3, 481–490. https://doi.org/10.1017/S0004972700035310
  13. 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
  14. R. Y. Sharp, Steps in Commutative Algebra, London Mathematical Society Student Texts, 19. Cambridge University Press, Cambridge, 1990.

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  2. TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS vol.51, pp.3, 2014, https://doi.org/10.4134/JKMS.2014.51.3.593
  3. An ideal based zero-divisor graph of a commutative semiring vol.44, pp.1, 2009, https://doi.org/10.3336/gm.44.1.07
  4. THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS vol.51, pp.1, 2014, https://doi.org/10.4134/JKMS.2014.51.1.189