On a graph of ideals of a commutative ring

Year 2019, Volume: 68 Issue: 2, 2283 - 2297, 01.08.2019

Mehdi Khoramdel Shahabaddin Ebrahimi Atani Saboura Dolati Pishhesari

https://doi.org/10.31801/cfsuasmas.534944

Cited By: 1

Abstract

In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J). In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which G(R) is planar, complete and complete r-partite. We show that, if (R,M) is a local Artinian ring, then G(R) is complete if and only if Soc(R) is simple. Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.

Keywords

Ideals, clique number, complete r-partite, planar property

References

• Akbari, S., Habibi, M., Majidinya, A., and Manaviyat, R., The inclusion ideal graph of rings, Communications in Algebra, 43 (2015), 2457--2465.
• Anderson, D. F., and Badawi, A., The total graph of a commutative ring, J. Algebra, 320 (2008), no. 7, 2706--2719.
• Anderson, D. F., and Badawi, A., Von Neumann Regular and Related Elements in Commutative Rings, Algebra Colloquium, (2012), Vol. 19, No. spec01 : pp. 1017-1040.
• Anderson, D. F., and Livingston, P. S., The zero-divisor graph of a commutative rings, J. Algebra, 217 (1999), 434--447.
• Atiyah, M. F., and Macdonald, I. G., Introduction to Commutative Algebra. Addison- Wesley Publishing Company, 1969.
• Beck, I., Coloring of commutative rings. J. Algebra, 116 (1988), 208--226.
• Bondy, J. A., Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008.
• Bruns, W., Herzog, J., Cohen-Macaulay Rings, Cambridge University Press, 1997.
• Camillo, V., Nicholson, W. K., Yousif, M. F., Ikeda-Nakayama Rings, Journal of Algebra, 226 (2000), 1001--1010.
• Chakrabarty, I., Ghosh, S., Mukherjee T. K., and Sen, M. K., Intersection graphs of ideals of rings, Discrete Math., 309 (2009) 5381--5392.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., The identity-summand graph of commutative semirings, J. Korean Math. Soc., 51 (2014), No. 1, pp. 189--202.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., Total graph of a commutative semiring with respect to identity-summand elements, J. Korean Math. Soc., 51 (3) (2014), 593--607.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., Total identity-summand graph of commutative semirings with respect to a co-ideal, J. Korean Math. Soc., 52(1) (2015), 159--176.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., A co-ideal based identity-summand graph of a commutative semiring, Commentationes Mathematicae Universitatis Carolinae, 56,3 (2015) 269--285.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., A graph associated to proper non-small ideals of a commutative ring, Comment.Math.Univ.Carolin., 58,1 (2017) 1--12.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., Ebrahimi Sarvandi, Z., Intersection graphs of co-ideals of semirings, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics , Volume 68, Number 1, (2019), Pages 840--851.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., and Sedghi Shanbeh Bazari, M., Total graph of a 0-distributive lattice, Categories and General Algebraic Structures with Applications, 9(1), (2018), 15--27.
• Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., and Sedghi Shanbeh Bazari, M., A semiprime filter-based identity-summand graph of lattices, Le Matematiche, 2 (2018), 297--318.
• Goodearl K. R., and Warfield, R. B., An Introduction to Noncommutative Noetherian Rings, 2nd edn. Cambridge University Press, Cambridge, 2004.
• Haynes, T. W., Hedetniemi S. T., and Slater P. J., (eds.), Fundamentals of Domination in graphs Marcel Dekker, Inc, New York, NY, 1998.
• Ikeda, M., and Nakayama, T., On some characteristic properties of quasi-Frobenius and regular rings, Proc. Amer. Math. Soc., 5 (1954), 15--19.
• Kaplansky, I., Dual rings, Ann. Math., 49 (1948), 689--701.
• Wisbauer, R., Foundations of Module and Ring Theory. Philadelphia: Gordon and Breach, 1991.
• Ye, M., Wu, TS., Comaximal ideal graphs of commutative rings. J Algebra Appl., (2012); 11: 1250114 (14 pages).