Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE TOTAL GRAPH OF A COMMUTATIVE RING WITH RESPECT TO PROPER IDEALS

  • Abbasi, Ahmad (Department of Pure Mathematics Faculty of Mathematical Sciences University of Guilan) ;
  • Habibi, Shokoofe (Department of Pure Mathematics Faculty of Mathematical Sciences University of Guilan)
  • Received : 2010.08.24
  • Published : 2012.01.01
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Abstract

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(${\Gamma}_I(R)$). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ${\in}$ R, the vertices x and y are adjacent if and only if x + y ${\in}$ S(I). The total graph of a commutative ring, that denoted by T(${\Gamma}(R)$), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ${\in}$ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, $T({\Gamma}_I(R))=T({\Gamma}(R))$; this is an important result on the definition.

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References

  1. S. Akbari, D. Kiani, F. Mohammadi, and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra 213 (2009), no. 12, 2224-2228. https://doi.org/10.1016/j.jpaa.2009.03.013
  2. D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706-2719. https://doi.org/10.1016/j.jalgebra.2008.06.028
  3. 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
  4. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Readig, MA, 1969.
  5. B. Bollobas, Graph Theory: An Introduction Course, Springer-Verlag, New York, 1979.
  6. A. Yousefian Darani, Primal and weakly primal submodules and related results, Ph. D. thesis, University of Guilan, Iran, 2008.
  7. L. Fuchs, On primal ideals, Proc. Amer. Math. Soc. 1 (1950), 1-6. https://doi.org/10.1090/S0002-9939-1950-0032584-8
  8. 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

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