Some properties of the essential annihilating-ideal graph of commutative rings

Document Type : Original paper

Authors

1 Department of mathematics, Aligarh Muslim University, Aligarh.

2 Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

Abstract

Let $\mathcal{S}$ be a commutative ring with unity and $A(\mathcal{S})$ denotes the set of annihilating-ideals of $\mathcal{S}$. The essential annihilating-ideal graph of $\mathcal{S}$, denoted by $\mathcal{EG}(\mathcal{S})$, is an undirected graph with $A^*(\mathcal{S})$ as the set of vertices and   for distinct $\mathcal{I}, \mathcal{J} \in A^*(\mathcal{S})$, $\mathcal{I} \sim \mathcal{J}$ is an edge if and only if $Ann(\mathcal{IJ}) \leq_e \mathcal{S}$. In this paper, we classify the Artinian rings $\mathcal{S}$ for which $\mathcal{EG}(\mathcal{S})$ is projective. We also discuss the coloring of $\mathcal{EG}(\mathcal{S})$. Moreover, we discuss the domination number of $\mathcal{EG}(\mathcal{S})$.

Keywords

Main Subjects

References

[1] A. Alilou and J. Amjadi, The sum-annihilating essential ideal graph of a commutative ring, Commun. Comb. Optim. 1 (2016), no. 2, 117–135.
[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.D. Anderson and P.S. Livingston, Coloring of commutative rings, J. Algebra 217 (1999), no. 2, 434–447.
[4] D.D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500–514.
[5] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.
[6] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226.
[7] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727–739.
[8] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 741–753.
[9] S. Krishnan and P. Subbulakshmi, Classification of rings with toroidal annihilating-ideal graph, Commun. Comb. Optim. 3 (2018), no. 2, 93–119.
[10] M. Nazim and N. ur Rehman, On the essential annihilating-ideal graph of commutative rings, Ars Math. Contemp. 22 (2022), no. 3, #P3.05.
[11] R. Nikandish and H.R. Maimani, Dominating sets of the annihilating-ideal graphs, Electron. Notes Discrete Math. 45 (2014), 17–22.
[12] K. Selvakumar and P. Subbulakshmi, On the crosscap of the annihilating-ideal graph of a commutative ring, Palestine J. Math. 7 (2018), no. 1, 151–160.
[13] 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, ID: 1750058.
[14] D.B. West, Introduction to Graph Theory, Prentice-Hall of India, New Delhi, 2001.
[15] A.T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam, 1973.
Communications in Combinatorics and Optimization
Volume 8, Issue 4
December 2023
Pages 715-724

Files

Share

How to cite

Statistics