On the (outer)planarity of the intersection graph of idealization

Articles in Press

Document Type : Original paper

Authors

1 Department of Pure Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran

2 Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

Let $R$ be a commutative ring with identity. The intersection graph of ideals of a ring $R$ is an undirected simple graph denoted by $\Gamma(R)$ whose vertices are in a one-to-one correspondence with nonzero proper ideals and two distinct vertices are joined by an edge if and only if the corresponding ideals of $R$ have a nonzero intersection. Let $M$ be a unitary nonzero $R$-module and let $R\ltimes M$ be the idealization of $M$ in $R$. In this paper, we provide a characterization of the planarity of the intersection graph of idealization. We then conclude that the intersection graph of

idealization is planar if and only if it is outerplanar.

Keywords

Main Subjects

References

[1] S. Akbari and R. Nikandish, Some results on the intersection graph of ideals of matrix algebras, Linear Multilinear Algebra 62 (2014), no. 2, 195–206. https://doi.org/10.1080/03081087.2013.769101
[2] S. Akbari, R. Nikandish, and M.J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013), no. 4, 1250200. https://doi.org/10.1142/S0219498812502003
[3] D. Anderson and M.R. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3–56. https://doi.org/10.1216/JCA-2009-1-1-3
[4] 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
[5] M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Boston, 1969.
[6] M. Axtell and J. Stickles, Zero-divisor graphs of idealizations, J. Pure Appl. Algebra 204 (2006), no. 2, 235–243.
https://doi.org/10.1016/j.jpaa.2005.04.004
[7] C. Bakkari, S.E. Kabbaj, and N. Mahdou, Trivial extensions defined by prüfer conditions, J. Pure Appl. Algebra 214 (2010), no. 1, 53–60. https://doi.org/10.1016/j.jpaa.2009.04.011
[8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
[9] N. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727–739. https://doi.org/10.1142/S0219498811004896
[10] I. Chakrabarty, S. Ghosh, T.K. Mukherjee, and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009), no. 17, 5381–5392. https://doi.org/10.1016/j.disc.2008.11.034
[11] J.A. Huckaba, Commutative Rings with Zero Divisors, M. Dekker, New York, 1988.
[12] A. Mahmoodi, A. Vahidi, R. Manaviyat, and R. Alipour, Intersection graph of idealizations, Collect. Math. 75 (2024), no. 3, 693–702. https://doi.org/10.1007/s13348-023-00407-7
[13] T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.
[14] M. Nagata, Local Rings, Interscience, New York, 1962.
[15] D.B. West, Introduction to Graph Theory, second ed., Prentice Hall, 2001.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 07 July 2026

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