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Year 2018, Volume: 5 Issue: 2, 85 - 99, 15.05.2018
https://doi.org/10.13069/jacodesmath.423751

Abstract

References

  • [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).

Some results on the comaximal ideal graph of a commutative ring

Year 2018, Volume: 5 Issue: 2, 85 - 99, 15.05.2018
https://doi.org/10.13069/jacodesmath.423751

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 $\mathscr{C}(R)$.

References

  • [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).
There are 18 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Subramanian Visweswaran This is me 0000-0002-4905-809X

Jaydeep Parejiya This is me 0000-0002-2072-2719

Publication Date May 15, 2018
Published in Issue Year 2018 Volume: 5 Issue: 2

Cite

APA Visweswaran, S., & Parejiya, 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. https://doi.org/10.13069/jacodesmath.423751
AMA Visweswaran S, Parejiya J. Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2018;5(2):85-99. doi:10.13069/jacodesmath.423751
Chicago Visweswaran, Subramanian, and Jaydeep Parejiya. “Some Results on the Comaximal Ideal Graph of a Commutative Ring”. Journal of Algebra Combinatorics Discrete Structures and Applications 5, no. 2 (May 2018): 85-99. https://doi.org/10.13069/jacodesmath.423751.
EndNote Visweswaran S, Parejiya J (May 1, 2018) Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications 5 2 85–99.
IEEE S. Visweswaran and J. Parejiya, “Some results on the comaximal ideal graph of a commutative ring”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 5, no. 2, pp. 85–99, 2018, doi: 10.13069/jacodesmath.423751.
ISNAD Visweswaran, Subramanian - Parejiya, 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.
JAMA Visweswaran S, Parejiya J. Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5:85–99.
MLA Visweswaran, Subramanian and Jaydeep Parejiya. “Some Results on the Comaximal Ideal Graph of a Commutative Ring”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 5, no. 2, 2018, pp. 85-99, doi:10.13069/jacodesmath.423751.
Vancouver Visweswaran S, Parejiya 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.

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