On the metric dimension of a total graph of non-zero annihilating ideals

Abachi, Nazi; Sahebi, Shervin

doi:10.2478/auom-2020-0031

Abstract

Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R − {0} such that Ir = (0). Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by Ω(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, J are joined if and only if I +J is also an annihilating ideal of R. In this paper, we study the metric dimension of Ω(R) and provide metric dimension formulas for Ω(R).

References

[1] D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447.10.1006/jabr.1998.7840
Search in Google Scholar Back to article
[2] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014), 108–121.10.1080/00927872.2012.707262
Search in Google Scholar Back to article
[3] A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra, 43 (2015), 43–50.10.1080/00927872.2014.897188
Search in Google Scholar Back to article
[4] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, (1969).
Search in Google Scholar Back to article
[5] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, (1997).10.1017/CBO9780511608681
Search in Google Scholar Back to article
[6] N. Ganesan, Properties of rings with a finite number of zero-divisors, Math. Ann., 157 (1964), 215–218.10.1007/BF01362435
Search in Google Scholar Back to article
[7] S. Pirzada, R. Raja, S. P. Redmond, Locating sets and numbers of graphs associated to commutative rings, J. Algebra Appl., 13:7 (2014), 1450047 (18pp).10.1142/S0219498814500479
Search in Google Scholar Back to article
[8] S. Pirzada, R. Raja, On the metric dimension of a zero-divisor graph, Comm. Algebra, 45:4 (2017), 1399–1408.10.1080/00927872.2016.1175602
Search in Google Scholar Back to article
[9] R. Raja, S. Pirzada, S. P. Redmond, On Locating numbers and codes of zero-divisor graphs associated with commutative rings, J. Algebra Appl., 15:1 (2016), 1650014 (22pp).10.1142/S0219498816500146
Search in Google Scholar Back to article
[10] F. Harary, R. A. Melter, On the metric dimension of a graph, Ara Combin., 2 (1976), 191–195.
Search in Google Scholar Back to article
[11] R. Nikandish, M.J. Nikmehr, M. Bakhtyiari, Coloring of the annihilator graph of a commutative ring, J. Algebra Appl., 15:6 (2016), 1650124 (13 pp).10.1142/S0219498816501243
Search in Google Scholar Back to article
[12] J. A. Huckaba, Commutative Rings with Zero Divisors, 2nd ed., Prentice Hall, Upper Saddle River, (1988).
Search in Google Scholar Back to article
[13] T. Y. Lam, A First Course in Non-Commutative Rings Springer-Verlag, New York, Inc, (1991).10.1007/978-1-4684-0406-7
Search in Google Scholar Back to article
[14] S. Visweswaram, H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative Ring, Discrete Mathematics, Algorithms and Applications, 6:4 (2014), 1450047 (22 pp).10.1142/S1793830914500475
Search in Google Scholar Back to article
[15] D.B. West, Introduction to Graph Theory, 2nd ed, Prentice Hall, Upper Saddle River, (2001).
Search in Google Scholar Back to article

On the metric dimension of a total graph of non-zero annihilating ideals

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