Some Remarks on the Annihilating-Ideal Graph of Commutative Ring with Respect to an Ideal

Document Type : Original Scientific Paper

Authors

‎Department of mathematics, ‎Faculty of science, ‎University of Qom‎, ‎ ‎I‎. ‎R‎. ‎Iran

Abstract

‎The graph $ AG ( R ) $ {of} a commutative ring $R$ with identity has an edge linking two unique vertices when the product of the vertices equals {the} zero ideal and its vertices are the nonzero annihilating ideals of $R$‎.
‎The annihilating-ideal graph with {respect to} an ideal $ ( I ) $, ‎which is {denoted} by $ AG_I ( R ) $‎, ‎has distinct vertices $ K $ and $ J $ that are adjacent if and only if $ KJ\subseteq I $‎. ‎Its vertices are $ \{K\mid KJ\subseteq I\ \text{for some ideal}\ J \ \text{and}\ K$‎, ‎$J \nsubseteq I‎, ‎K\ \text{is a ideal of}\ R\} $‎. ‎The study of the two graphs $ AG_I ( R ) $ and $ AG(R/I) $ and {extending certain} prior findings are two main objectives of this research‎. ‎This studys {among other things‎, ‎the} findings {of this study reveal}‎
‎that $ AG_I ( R ) $ is bipartite if and only if $ AG_I ( R ) $ is triangle-free‎.

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References

[1] R. Y. Sharp, Step in Commutative Algebra, London Mathematical Society Student Texts (51), Cambridge University Press, Cambridge, 2000.
[2] B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, M. Dekker, Inc., New York, 1974.
[3] 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.
[4] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Commun. Algebra 31 (2003) 4425 - 4443, https://doi.org/10.1081/AGB-120022801.
[5] M. Behboodi and Z. Rakeei, The annihilating - ideal graph of a commutative rings I, J. Algebra Appl. 10 (2011) 727 - 739, https://doi.org/10.1142/S0219498811004896.
[6] F. Aliniaeifard, M. Behboodi, E. Mehdi-Nezhad and A. M. Rahimi, The annihilating-ideal graph of a commutative ring with respect to an ideal, Comm. Algebra 42 (2014) 2269 - 2284, https://doi.org/10.1080/00927872.2012.753606.
[7] S. Visweswaran and H. D. Patel, Some results on the complement of the annihilating ideal graph of a commutative ring, J. Algebra Appl. 14 (2015) p. 1550099, https://doi.org/10.1142/S0219498815500991.
[8] S. Visweswaran and H. D. Patel, On the clique number of the complement of the annihilating ideal graph of a commutative ring, Beitr. Algebra Geom. 57 (2016) 307 - 320, https://doi.org/10.1007/s13366-015-0247-5.
[9] S. Visweswaran and A. Parmar, Some results on a spanning subgraph of the complement of the annihilating-ideal graph of a commutative reduced ring, Discrete Math. Algorithms Appl. 11 (2019) p. 1950012, https://doi.org/10.1142/S1793830919500125.
[10] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The classification of the annihilating - ideal graph of a commutative ring, Algebra Colloq. 21 (2014) 249 - 256, https://doi.org/10.1142/S1005386714000200.
[11] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, On the coloring of the annihilating - ideal graph of a commutative ring, Discrete Math. 312 (2012) 2620 - 2626, https://doi.org/10.1016/j.disc.2011.10.020.
[12] S. Visweswaran, P. T. Lalchandani, When is the annihilating ideal graph of a zero-dimensional semiquasilocal commutative ring planar? Nonquasilocal case, Boll. Unione Mat. Ital. 9 (2016) 453 - 468, https://doi.org/10.1007/s40574-016-0061-5.
Mathematics Interdisciplinary Research
Volume 9, Issue 1
March 2024
Pages 111-129

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