Some results on the comaximal ideal graph of a commutative ring

Abstract

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let $R$ be a ring such that $R$ admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of $R$, denoted by $\mathscr{C}(R)$ is an undirected simple graph whose vertex set is the set of all proper ideals $I$ of $R$ such that $I\not\subseteq J(R)$, where $J(R)$ is the Jacobson radical of $R$ and distinct vertices $I_{1}$, $I_{2}$ are joined by an edge in $\mathscr{C}(R)$ if and only if $I_{1} + I_{2} = R$. In Section 2 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is planar. In Section 3 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is a split graph. In Section 4 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is complemented and moreover, we determine the $S$-vertices of $\mathscr{C}(R)$.

Keywords

• Comaximal ideal graph,Special principal ideal ring,Planar graph,Split graph,Complement of a vertex in a graph

References

1. [1] D. F. Anderson, P. S. Livingston, The zero–divisor graph of a commutative ring, J. Algebra 217(2) (1999) 434–447.
2. [2] D. F. Anderson, R. Levy, J. Shapiro, Zero–divisor graphs, von Neumann regular rings, and Boolean Algebras, J. Pure Appl. Algebra 180(3) (2003) 221–241.
3. [3] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, Reading, Massachusetts, 1969.
4. [4] R. Balakrishnan, K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, 2000.
5. [5] M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739.
6. [6] M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753.
7. [7] N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India Private Limited, New Delhi, 1994.
8. [8] R. Gilmer, Multiplicative Ideal Theory, Marcel–Dekker, New York, 1972.

9. [9] M. I. Jinnah, S. C. Mathew, When is the comaximal graph split?, Comm. Algebra 40(7) (2012) 2400–2404.
10. [10] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
11. [11] R. Levy, J. Shapiro, The zero–divisor graph of von Neumann regular rings, Comm. Algebra 30(2) (2002) 745–750.
12. [12] H. R. Maimani, M. Salimi, A. Sattari, S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319(4) (2008) 1801–1808.
13. [13] S. M. Moconja, Z. Z. Petrovic, On the structure of comaximal graphs of commutative rings with identity, Bull. Aust. Math. Soc. 83(1) (2011) 11–21.
14. [14] A. M. Rahimi, Smarandache vertices of the graphs associated to the commutative rings, Comm. Algebra 41(5) (2013) 1989–2004.
15. [15] K. Samei, On the comaximal graph of a commutative ring, Canad. Math. Bull. 57(2) (2014) 413–423.
16. [16] P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1) (1995) 124–127.
17. [17] S. Visweswaran, P. Sarman, On the planarity of a graph associated to a commutative ring and on the planarity of its complement, S~ao Paulo J. Math. Sci. 11(2) (2017) 405–429.
18. [18] M. Ye, T. Wu, Co–maximal ideal graphs of commutative rings, J. Algebra Appl. 11(6) (2012) Article ID. 1250114 (14 pages).