On intersection minimal ideal graph of a ring

Document Type : Research Paper

Authors

Department of Mathematics, Cotton University, Guwahati-781001, India

Abstract

For a ring $R$, the intersection minimal ideal graph, denoted by $ \wedge(R) $, is a simple undirected graph whose vertices are proper non-zero (right) ideals of $R$ and any two distinct vertices $I_{1}$ and $I_{2}$ are adjacent if and only if $ I_{1} \cap I_{2}$ is a minimal ideal of $R$. In this article, we explore connectedness, clique number, split character, planarity, independence number, domination number of $\wedge(R)$.
 

Keywords

References

[1] F. H. Abdulqadr, Maximal ideal graph of commutative rings, Iraqi J. Sci., 61 No. 8 (2020) 2070–2076.
[2] S. Akbari, R. Nikandish, and M. J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Its Appl., 12 No. 04 (2013) 1250200.
[3] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2020) 2706-2719.
[4] S. E. Atani, S. Hesari and M. Khoramdel, A graph associated to proper non-small ideals of a commutative ring, Comment. Math. Univ. Carol., 58 No. 1 1-12.
[5] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Second edition, Addison-Wesley, London, 1969.
[6] R. Balkrishnan and K. Ranganathan, A Text Book of Graph Theory, Springer-verlag New York, Inc., 2008.
[7] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208-226.
[8] I. Chakrabarty and J. V. Kureethara, A survey on the intersection graphs of ideals of rings, Commun. comb. optim., 7 No. 2 (2022) 121-167.
[9] I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309 No. 17 (2009) 5381-5392.
[10] K. R. Goodearl, Ring Theory, Second edition, Marcel Dekker, 1976.
[11] F. Harary, Graph Theory, Addison-Wesley Publishing Company, New York, 1969.
[12] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.
[13] J. A. Huckaba, Commutative Rings with Zero-Divisors, Marcel-Dekker, New York, Basel, 1988.
[14] I. Kaplansky, Commutative Rings, Revised edition, University of Chicago Press, Chicago, 1974.
[15] F. Kasch, Modules and Rings, Academic Press Inc., London Ltd., 1982.
[16] T. Lam, Lectures on Nodules and Rings, Graduate Texts in Mathematics, Volume 189, Springer, New York, 1999.
[17] J. Lambeck, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, Toronto, London, 1966.
[18] K. K. Rajkhowa and H. K. Saikia, Center of intersection graph of submodules of a module, AKCE Int. J. Graphs Comb., 16 No. 2 (2019) 198-203.
[19] K. K. Rajkhowa and H. K. Saikia, Prime intersection graph of ideals of a ring, Proc. Math. Sci., 130 (2020) 0973-7685.
[20] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 No. 1 (1995) 124-127.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 05 June 2024

Files

Share

How to cite

Statistics