Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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THE TOTAL GRAPH OF NON-ZERO ANNIHILATING IDEALS OF A COMMUTATIVE RING

  • Alibemani, Abolfazl (Faculty of Mathematical Sciences Shahrood University of Technology) ;
  • Hashemi, Ebrahim (Faculty of Mathematical Sciences Shahrood University of Technology)
  • Received : 2017.05.29
  • Accepted : 2017.11.02
  • Published : 2018.04.30
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Abstract

Assume that R is a commutative ring with non-zero identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists a non-zero element $a{\in}R$ such that Ia = 0. S. Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by ${\Omega}(R)$, as the graph with the vertex-set $A(R)^*$, the set of all non-zero annihilating ideals of R, and two distinct vertices I and J are adjacent if I + J is an annihilating ideal. In this paper, we study the relations between the diameters of ${\Omega}(R)$ and ${\Omega}(R[x])$. Also, we study the relations between the diameters of ${\Omega}(R)$ and ${\Omega}(R[[x]])$, whenever R is a Noetherian ring. In addition, we investigate the relations between the diameters of this graph and the zero-divisor graph. Moreover, we study some combinatorial properties of ${\Omega}(R)$ such as domination number and independence number. Furthermore, we study the complement of this graph.

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References

  1. 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
  2. M. Axtell, J. Coykendall, and J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra 33 (2005), no. 6, 2043-2050. https://doi.org/10.1081/AGB-200063357
  3. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  4. D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (1971), 427-433. https://doi.org/10.1090/S0002-9939-1971-0271100-6
  5. J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
  6. I. Kaplansky, Commutative Rings, Revised edition, The University of Chicago Press, Chicago, IL, 1974.
  7. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.
  8. 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
  9. R. Y. Sharp, Steps in Commutative Algebra, Second edition, London Mathematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
  10. S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihi- lating ideals of a commutative ring, Discrete Math. Algorithms Appl. 6 (2014), no. 4, 1450047, 22 pp. https://doi.org/10.1142/S1793830914500475
  11. S. Visweswaran and P. Sarman, On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 8 (2016), no. 3, 1650043, 22 pp. https://doi.org/10.1142/S1793830916500439
  12. D. B. West, Introduction to Graph Theory, Second Edition, Prentice Hall, Upper Saddle River, 2001.