Comaximal Intersection Graph of Ideals of Rings

Document Type : Research Paper

Authors

1 Department of Mathematics Cotton University Guwahati-781001, INDIA

2 Department of Mathematics Bineswar Brahma Engineering College Kokrajhar-783370, INDIA

3 Department of Mathematics Shivaji College University of Delhi New Delhi-110027, INDIA

Abstract

The comaximal intersection graph $CI(R)$ of ideals of a ring $R$ is an undirected graph whose vertex set is the collection of all non-trivial (left) ideals of $R$ and any two vertices $I$ and  $J$ are adjacent if and only if $I+J=R$ and $I\cap J\neq0$. We study the connectedness of $CI(R)$. We also discuss  independence number, clique number, domination number, chromatic number of $CI(R)$.
 

Keywords

References

1. G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaveisi, Minimal prime ideals and cycles in annihilating-ideal graphs, Rocky Mountain J. Math., (5) 43 (2013), 1415-1425.
2. G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, , M. J. Nikmehr, and F. Shaveisi, The Classi cation of the Annihilating-Ideal Graphs of Commutative Rings, Algebra Colloq. (2) 21 (2014), 249-256.
3. S. Akbari, R. Nikandish, and M. J. Nikmehr, Some Results on the Intersection Graphs of Ideals of Rings, J. Algebra Appl. (4) 12 (2013), 1250200 (13 pages).
4. D. F. Anderson, M. C. Axtell, and J. A. Jr. Stickles, Zero-divisor graphs in commutative rings. In: Fontana, M., Kabbaj, S.-E., Olberding, B., Swanson, I., eds., Commutative Algebra, Noetherian and Non-Noetherian Perspectives,
New York: Springer-Verlag, (2011), 23-45.
5. D. F. Anderson and P. S. Livingston, The Zero-Divisor Graph of a Commutative Ring, J. Algebra, (2) 217 (1999), 434-447.
6. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra. Reading, Massachusetts: Addison-Wesley , 1969.
7. R. Balkrishnan and K. Ranganathan, A Text Book of Graph Theory. New York: Springer, 2012.
8. B. Barman and K. K. Rajkhowa, Non-comaximal graph of ideals of a ring, Proc. Indian Acad. Sci. (Math. Sci.), (5) 129 (2019), 1-8.
9. I. Beck, Coloring of commutative rings, J. Algebra, (1) 116 (1988), 208-226.
10. M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. (4) 10 (2011), 727-739.
11. M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings
II, J. Algebra Appl. (4) 10 (2011), 741-753.
12. I. Chakrabarty, S. Ghosh, T. K. Mukherjee, and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math. (17) 309 (2009), 5381-5392.
13. E. J. Cockayne, S. T. Hedetniemi, Towards a Theory of Domination in Graphs, Networks, (3) 7 (1977), 247{261.
14. J. A. Gallian, Contemporary Abstract Algebra. New Delhi, India : Narosa Publishing House , 1999.
15. K. R. Goodearl, Ring Theory, Nonsingular Rings and Modules. New York and Basel : MARCEL DEKKER, 1976.
16. F. Harary, Graph Theory. Reading, Massachusetts: Addison-Wesley, 1969.
17. H. R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, (4) 319 (2008), 1801-1808.
18. K. K. Rajkhowa and H. K. Saikia Prime intersection graph of ideals of a ring,
Proc. Indian Acad. Sci. (Math. Sci.), (17) 130 (2020), 1-12.
19. E. Sampathkumar, The Global Domination Number of A Graph J. Math. Phy.Sci., (5) 23 (1989), 377-385.
20. P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, (1) 176 (1995), 124-127.
21. S. Visweswaran and P. Vadhel, Some results on a subgraph of the intersection graph of ideals of a commutative ring, J. Algebra Relat. Topics, (2) 6(2018), 35-61.
22. M. Ye and TS. Wu, Comaximal ideal graphs of commutative rings, J. Algebra Appl. (6) 11 (2012), 1250114.
Journal of Algebra and Related Topics
Volume 12, Issue 1
July 2024
Pages 147-157

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