Some Remarks on $(\operatorname{INC}(R))^{c}$

Document Type : Research Paper

Authors

1 Department of Applied Sciences, RK University, Rajkot-360020, Gujarat, India.

2 Department of Mathematics, Government Polytechnic, Rajkot-360003, Gujarat, India.

Abstract

Let $R$ be a commutative ring with identity $1 \neq 0$ which admits atleast two maximal ideals. In this article, we have studied simple, undirected graph $(\operatorname{INC}(R))^{c}$ whose vertex set is the set of all proper ideals which are not contained in $J(R)$ and two distinct vertices $I_{1}$ and $I_{2}$ are joined by an edge in $(\operatorname{INC}(R))^{c}$ if and only if $I_{1} \subseteq I_{2}$ or $I_{2} \subseteq I_{1}$. In this article, we have studied some interesting properties of $(\operatorname{INC}(R))^{c}$.

Keywords

References

[1] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaiveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math., 312 (2012) 2620-2625.
[2] S. Akbari, B. Miraftab and R. Nikandish, Co-maximal graph of subgroups of groups, Can. Math. Bull., 60 No. 1 (2017) 12-25.
[3] S. Akbari, R. Nikandish and M. J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl., 12 No. 04 (2013) 1250200.
[4] D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings and Boolean Algebras, J. Pure Appl. Algebra, 180 (2003) 221-241.
[5] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999) 434-447.
[6] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969.
[7] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, 2000.
[8] I. Beck, Coloring of commutative rings, J. Algebra, 116 No. 1 (1988) 208-226.
[9] M. Behboodi and Z. Rakeei, The annihilating-ideal graphs of commutative rings I, J. Algebra Appl., 10 No. 4 (2011) 727-739.
[10] M. Behboodi and Z. Rakeei, The annihilating-ideal graphs of commutative rings II, J. Algebra Appl., 10 No. 4 (2011) 741-753.
[11] N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India Private Limited, New Delhi, 1994.
[12] M. I. Jinnah and S. C. Mathew, When is the comaximal graph split?, Comm. Algebra, 40 No. 7 (2012) 2400-2404.
[13] H. R. Maimani, M. Salimi, A. Sattari and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319 No. 4 (2008) 1801-1808.
[14] B. Miraftab, R. Nikandish, Co-maximal graphs of two generator groups, J. Algebra its Appl., 18 No. 04 (2019) 1950068.
[15] S. M. Moconja and Z. Z. Petrovic, On the structure of comaximal graphs of commutative rings with identity, Bull. Aust. Math. Soc., 83 (2011) 11-21.
[16] K. Nazzal and M. Ghanem, On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo, Int. J. Comb, 2012 (2012).
[17] R. Nikandish and H. R. Maimani, Dominating sets of the annihilating-ideal graphs, Electron. Notes Discrete Math., 45 (2014) 17-22.
[18] K. Samei, On the comaximal graph of a commutative ring, Can. Math. Bull., 57 No. 2 (2014) 413-423.
[19] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995) 124-127.
[20] A. Sharma and A. Gaur, Line Graphs associated to the Maximal graph, J. Algebra Relat. Top., 3 No. 1 (2015) 1-11.
[21] S. Visweswaran and J. Parejiya, Annihilating -ideal graphs with independence number at most four, Cogent Math., 3 No. 1 (2016).
[22] S. Vishweswaran and J. Parejiya, Some results on a supergraph of the comaximal ideal graph of a commutative ring, Commun. Comb. Optim., 3 No. 2 (2018) 1-22.
[23] S. Visweswaran and J. Parejiya, When is the complement of the comaximal graph of a commutative ring planar?, ISRN Algebra, 8 No. 2014 (2014).
[24] S. Visweswaran and H. D. Patel, Some results on the complement of the annihilating ideal graph of a commutative ring, J. Algebra Appl., 14 No. 7 (2015), 1550099.
[25] H. J. Wang, Graphs associated to Co-maximal ideals of commutative rings, J. Algebra, 320 No. 7 (2008) 2917-2933.
[26] M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra Appl., 11 No. 6 (2012), 1250114.
Algebraic Structures and Their Applications
Volume 9, Issue 2
August 2022
Pages 181-198

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