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2015 | 35 | 2 | 195-204
Tytuł artykułu

On the connectivity of the annihilating-ideal graphs

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EN
Abstrakty
EN
Let R be a commutative ring with identity and 𝔸*(R) the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸*(R) and two distinct vertices I₁ and I₂ are adjacent if and only if I₁I₂ = (0). In this paper, we examine the presence of cut vertices and cut sets in the annihilating-ideal graph of a commutative Artinian ring and provide a partial classification of the rings in which they appear. Using this, we obtain the vertex connectivity of some annihilating-ideal graphs.
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Twórcy
  • Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India
  • Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India
Bibliografia
  • [1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434-447. doi: doi:10.1006/jabr.1998.7840
  • [2] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company (1969).
  • [3] M. Axtell, N. Baeth and J. Stickles, Cut vertices in zero-divisor graph of finite commutative rings, Comm. Algebra 39 (2011), 2179-2188. doi: 10.1080/00927872.2010.488681
  • [4] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (4) (2011), 727-739. doi: 10.1142/S0219498811004896
  • [5] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), 741-753. doi: 10.1142/S0219498811004902
  • [6] G. Chartrand and L. Lesniak, Graphs and Digraphs (Wadsworth and Brooks/Cole, Monterey, CA (1986).
  • [7] B. Cote, C. Ewing, M. Huhn, C.M. Plaut and D. Weber, Cut sets in zero-divisor graphs of finite commutative rings, Comm. Algebra 39 (2011), 2849-2861. doi: 10.1080/00927872.2010.489534
  • [8] I. Kaplansky, Commutative Rings, rev. ed. University of Chicago Press Chicago (1974).
  • [9] S.P. Redmond, Central sets and radii of the zero-divisor graphs of commutative rings, Comm. Algebra 34 (2006), 2389-2401. doi: 10.1080/00927870600649103
  • [10] T. Tamizh Chelvam and K. Selvakumar, Central sets in the annihilating-ideal graph of commutative rings, J. Combin. Math. Combin. Comput. 88 (2014), 277-288.
  • [11] A.T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam (1973).
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1241
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