| | | |

## Some results on the comaximal ideal graph of a commutative ring

#### Subramanian VİSWESWARAN [1] , Jaydeep PAREJİYA [2]

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 $\mathscr{C}(R)$.
Comaximal ideal graph, Special principal ideal ring, Planar graph, Split graph, Complement of a vertex in a graph
• [1] D. F. Anderson, P. S. Livingston, The zero–divisor graph of a commutative ring, J. Algebra 217(2) (1999) 434–447.
• [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] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, Reading, Massachusetts, 1969.
• [4] R. Balakrishnan, K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, 2000.
• [5] M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739.
• [6] M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753.
• [7] N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India Private Limited, New Delhi, 1994.
• [8] R. Gilmer, Multiplicative Ideal Theory, Marcel–Dekker, New York, 1972.
• [9] M. I. Jinnah, S. C. Mathew, When is the comaximal graph split?, Comm. Algebra 40(7) (2012) 2400–2404.
• [10] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
• [11] R. Levy, J. Shapiro, The zero–divisor graph of von Neumann regular rings, Comm. Algebra 30(2) (2002) 745–750.
• [12] H. R. Maimani, M. Salimi, A. Sattari, S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319(4) (2008) 1801–1808.
• [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] A. M. Rahimi, Smarandache vertices of the graphs associated to the commutative rings, Comm. Algebra 41(5) (2013) 1989–2004.
• [15] K. Samei, On the comaximal graph of a commutative ring, Canad. Math. Bull. 57(2) (2014) 413–423.
• [16] P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1) (1995) 124–127.
• [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] M. Ye, T. Wu, Co–maximal ideal graphs of commutative rings, J. Algebra Appl. 11(6) (2012) Article ID. 1250114 (14 pages).
Primary Language en Engineering Articles Orcid: 0000-0002-4905-809XAuthor: Subramanian VİSWESWARAN (Primary Author) Orcid: 0000-0002-2072-2719Author: Jaydeep PAREJİYA Publication Date : May 15, 2018
 Bibtex @research article { jacodesmath423751, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2018}, volume = {5}, pages = {85 - 99}, doi = {10.13069/jacodesmath.423751}, title = {Some results on the comaximal ideal graph of a commutative ring}, key = {cite}, author = {Vi̇sweswaran, Subramanian and Pareji̇ya, Jaydeep} } APA Vi̇sweswaran, S , Pareji̇ya, J . (2018). Some results on the comaximal ideal graph of a commutative ring . Journal of Algebra Combinatorics Discrete Structures and Applications , 5 (2) , 85-99 . DOI: 10.13069/jacodesmath.423751 MLA Vi̇sweswaran, S , Pareji̇ya, J . "Some results on the comaximal ideal graph of a commutative ring" . Journal of Algebra Combinatorics Discrete Structures and Applications 5 (2018 ): 85-99 Chicago Vi̇sweswaran, S , Pareji̇ya, J . "Some results on the comaximal ideal graph of a commutative ring". Journal of Algebra Combinatorics Discrete Structures and Applications 5 (2018 ): 85-99 RIS TY - JOUR T1 - Some results on the comaximal ideal graph of a commutative ring AU - Subramanian Vi̇sweswaran , Jaydeep Pareji̇ya Y1 - 2018 PY - 2018 N1 - doi: 10.13069/jacodesmath.423751 DO - 10.13069/jacodesmath.423751 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 85 EP - 99 VL - 5 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.423751 UR - https://doi.org/10.13069/jacodesmath.423751 Y2 - 2018 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Some results on the comaximal ideal graph of a commutative ring %A Subramanian Vi̇sweswaran , Jaydeep Pareji̇ya %T Some results on the comaximal ideal graph of a commutative ring %D 2018 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 5 %N 2 %R doi: 10.13069/jacodesmath.423751 %U 10.13069/jacodesmath.423751 ISNAD Vi̇sweswaran, Subramanian , Pareji̇ya, Jaydeep . "Some results on the comaximal ideal graph of a commutative ring". Journal of Algebra Combinatorics Discrete Structures and Applications 5 / 2 (May 2018): 85-99 . https://doi.org/10.13069/jacodesmath.423751 AMA Vi̇sweswaran S , Pareji̇ya J . Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018; 5(2): 85-99. Vancouver Vi̇sweswaran S , Pareji̇ya J . Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018; 5(2): 85-99.

Authors of the Article
[1]
[2]