On perfectness of annihilating-ideal graph of $\mathbb{Z}_n$

Document Type : Original paper


1 Presidency University, Kolkata

2 Department of Mathematics, Presidency University, Kolkata


The annihilating-ideal graph of a commutative ring $R$ with unity is defined as the graph $AG(R)$ whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ = 0$. Nikandish et.al. proved that $AG(\mathbb{Z}_n)$ is weakly perfect. In this short paper, we characterize $n$ for which $AG(\mathbb{Z}_n)$ is perfect.


Main Subjects

[1] C. AbdIoglu, E.Y. CelIkel, and A. Das, The armendariz graph of a ring, Discuss. Math. Gen. Algebra Appl. 38 (2018), no. 2, 189–196.
[2] J. Amjadi, R. Khoeilar, and A. Alilou, The annihilator-inclusion ideal graph of a commutative ring, Commun. Comb. Optim. 6 (2021), no. 2, 231–248.
[3] D.F. Anderson, M.C. Axtell, and J.A. Stickles, Jr., Zero-divisor graphs in commutative rings, Commutative Algebra: Noetherian and Non-Noetherian Perspectives (M. Fontana, S.-E. Kabbaj, B. Olberding, and I. Swanson, eds.), Springer,
2011, pp. 23–45.
[4] D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706–2719.
[5] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
[6] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42 (2014), no. 1, 108–121.
[7] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra . Appl. 10 (2011), no. 4, 727–739.
[8] B. Bose and A. Das, Graph theoretic representation of rings of continuous functions, Filomat 34 (2020), no. 10, 3417–3428.
[9] 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.
[10] I. Chakrabarty and J.V. Kureethara, A survey on the intersection graphs of ideals of rings, Commun. Comb. Optim. (in press).
[11] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Annals of Math. 164 (2006), no. 1, 51–229.
[12] A. Das, On perfectness of intersection graph of ideals of Zn, Discuss. Math. Gen. Algebra Appl. 37 (2017), no. 2, 119–126.
[13] M. Ebrahimi, The character graph of a finite group is perfect, Bull. Aust. Math. Soc. 104 (2021), no. 1, 127–131.
[14] H.R. Maimani, M.R. Pournaki, A. Tehranian, and S. Yassemi, Graphs attached to rings revisited, Arab. J. Sci. Eng. 36 (2011), no. 6, 997–1011.
[15] R. Nikandish, H.R. Maimani, and H. Izanloo, The annihilating-ideal graph of Zn is weakly perfect, Contrib. Discrete Math. 11 (2016), no. 1, 16–21.