On the genus of annihilator intersection graph of commutative rings

Document Type : Research Paper

Authors

1 Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India.

2 Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, India.

Abstract

Let $R$ be a commutative ring with unity and $A(R)$ be the set of annihilating-ideals of $R$. The annihilator intersection graph of $R$, represented by $AIG(R)$, is an undirected graph with $A(R)^*$ as the vertex set and $\mathfrak{M} \sim \mathfrak{N}$ is an edge of $AIG(R)$ if and only if $Ann(\mathfrak{M}\mathfrak{N}) \neq Ann(\mathfrak{M}) \cap Ann(\mathfrak{N})$, for distinct vertices $\mathfrak{M}$ and $\mathfrak{N}$ of $AIG(R)$. In this paper, we first defined finite commutative rings whose annihilator intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator intersection graph were characterized.

Keywords

References

[1] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999) 434-447.
[2] D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra, 159 No. 2 (1993) 500-514.
[3] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.
[4] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208-226.
[5] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative ring I, J. Algebra Appl., 10 (2011) 727-739.
[6] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative ring II, J. Algebra Appl., 10 (2011) 741-753.
[7] M. Nazim and N. Rehman, On the essential annihilating-ideal graph of commutative rings, Ars Math. Contemp., 22 No. 3 (2022) P3-05.
[8] M. Nazim, N. Rehman and S. A. Mir, Some properties of the essential annihilating-ideal graph of commutative rings, Commu. Comb. Opt., (2022) 1-10. DOI: 10.22049/CCO.2022.27827.1365
[9] N. Rehman, M. Nazim and K. Selvakumar, On the planarity, genus and crosscap of new extension of zero-divisor graph of commutative rings, AKCE Int. J. Graphs Comb., 19 No. 1 (2022) 61-68.
[10] M. Vafaei, A. Tehranian and R. Nikandish, On the annihilator intersection graph of a commutative ring, Italian J. Pure App. Mtah., 38 (2017) 531-541.
[11] D. B. West, Introduction to Graph Theory, Second edition, Prentice-Hall of India, New Delhi, 2002.
[12] A. T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam, 1973.
Algebraic Structures and Their Applications
Volume 11, Issue 1
February 2024
Pages 25-36

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