Some properties of comaximal ideal graph of a commutative ring

Document Type: Research Paper

Authors

Islamic Azad University, Central Tehran Branch

Abstract

Let $R$ be a commutative ring with identity‎. ‎We use‎ ‎$\varphi (R)$ to denote the comaximal ideal graph‎. ‎The vertices‎ ‎of $\varphi (R)$ are proper ideals of R which are not contained‎ ‎in the Jacobson radical of $R$‎, ‎and two vertices $I$ and $J$ are‎ ‎adjacent if and only if $I‎ + ‎J = R$‎. ‎In this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎associated to $\varphi (R)$‎.

Keywords

Main Subjects


[1] M. Azadi, Z. Jafari and Ch. Eslahchi, On the Comaximal ideal graph of a commutative ring, Turkish J. Math., 40 (2016) 905-913.

[2] R. Diestel, Graph Theory, Graduate Texts in Mathematics, 173, Springer-Verlag, New York, 2000.

[3] F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969.

[4] H. R. Maimani, M. Salimi, A. Sattari and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319 (2008) 1801-1808.

[5] J. Sedlacek, Some properties of interchange graphs, Theory of graphs and its applications, Academic Press, 1964 145-150.

[6] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995) 124-127.

[7] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Second edition, Cambridge University Press, Cambridge, 2001.

[8] W. Weisstein, Line Graph, From MathWorld-A Wolfrom WebResource.

[9] M. Ye and T. S. Wu, Comaximal ideal graphs of Commutative Rings, Journal of Algebra and Its Applications, 11 no. 6 (2012) pp. 14.