The annihilator-inclusion Ideal graph of a commutative ring

Document Type : Original paper

Authors

1 Azarbaijan Shahid Madani University

2 Jabir Ibn Hayyan research center

Abstract

Let R be a commutative ring with non-zero identity.   The  annihilator-inclusion ideal graph of R, denoted by ξR, is a graph whose vertex set is the of all non-zero proper ideals of R  and two distinct vertices I and J are adjacent if and only if either Ann(I)⊆J or Ann(J)⊆I.  The purpose of this paper is to provide some basic properties of the graph ξR. In particular, shows that ξR is a connected graph with diameter at most three, and has girth 3 or ∞.   Furthermore,  is  determined all isomorphic classes of non-local Artinian  rings whose annihilator-inclusion ideal graphs have genus zero or one.

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References

[1] S. Akbari, A. Alilou, J. Amjadi, and S.M. Sheikholeslami, The co-annihilating ideal graphs of commutative rings, Canadian Math. Bull. 60 (2017), 3–11.
[2] A. Alilou and J. Amjadi, The sum-annihilating essential ideal graph of a commutative ring, Commun. Comb. Optim. 1 (2016), no. 2, 117–135.
[3] J. Amjadi, The essential ideal graph of a commutative ring, Asian. Eur. J. Math. 11 (2018), no. 4, ID: 1850058.
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
[5] D.F. Anderson and S.B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007), no. 2,   543-550.
[6] J. Battle, F. Harary, and Y. Kodama, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), no. 6, 565–568.
[7] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2001), no. 4, 741–753.
[8] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727–739.
[9] K. Kuratowski, Sur le problèm des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283.
[10] H.R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319 (2008), no. 4, 1801–1808.
[11] G. Ringel, Das gescblecht des vollstandingen paaren graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139–150.
[12] G. Ringel and J.W.T. Youngs, Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 438–445.
[13] K. Selvakumar and Subbulakshmi, Classification of rings with toroidal annihilating ideal graph, Commun. Comb. Optim. 3 (2018), no. 2, 1–27.
[14] K. Selvakumar, P. Subbulakshmi, and J. Amjadi, On the genus of the graph associated to a commutative ring, Discrete Math. Algorithms Appl. 9 (2017), no. 5, 1750058 (11 pages).
[15] S. Visweswaran and P. Sarman, On the planarity of a graph associated to a commutative ring and on the planarity of its complement, São Paulo J. Math. Sci. 11 (2017), no. 2, 405–429.
[16] D.B. West, Introduction to Graph Theory, Prentice Hall, USA, 2001.
Communications in Combinatorics and Optimization
Volume 6, Issue 2
December 2021
Pages 231-248

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