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A GENERALIZED IDEAL BASED-ZERO DIVISOR GRAPHS OF NEAR-RINGS

  • Published : 2009.04.30
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Abstract

In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by $\widehat{{\Gamma}_I(N)}$. It is shown that if I is a completely reflexive ideal of N, then every two vertices in $\widehat{{\Gamma}_I(N)}$ are connected by a path of length at most 3, and if $\widehat{{\Gamma}_I(N)}$ contains a cycle, then the core K of $\widehat{{\Gamma}_I(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{{\Gamma}_I(N)}$ is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.

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References

  1. D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500–514 https://doi.org/10.1006/jabr.1993.1171
  2. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447 https://doi.org/10.1006/jabr.1998.7840
  3. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208–226 https://doi.org/10.1016/0021-8693(88)90202-5
  4. M. Behboodi and R. Beyranvand, Strong zero-divisor graphs of non-commutative rings, International Journal of Algebra 2 (2007), no. 1, 25–44
  5. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, Amsterdam, 1976
  6. A. Cannon, K. Neuerburg, and S. P. Redmond, Zero-divisor graph of anear-rings and semigroups, in: H. Kiechle, A. Kreuzer, M.J.Thomsen (Eds), Nearrings and Nearfields, Springer, Dordrecht, The Netherlands, 2005, 189–200
  7. F. R. DeMeyer, T. McKenzie, and K. Schneider, The zero-divisor graph of a commutative semigroups, Semigroup Forum 65 (2002), 206–214 https://doi.org/10.1007/s002330010128
  8. P. Dheena and B. Elavarasan, An ideal based-zero-divisor graph of 2-primal near-rings, (Submitted)
  9. S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533–3558
  10. G. Pilz, Near-Rings, North-Holland, Amsterdam, 1983
  11. 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

Cited by

  1. On generalized zero divisor graph of a poset vol.161, pp.10-11, 2013, https://doi.org/10.1016/j.dam.2012.12.019
  2. On zero-divisors of near-rings of polynomials pp.1727-933X, 2019, https://doi.org/10.2989/16073606.2018.1455070