A note on a graph related to the comaximal ideal graph of a commutative ring

Document Type : Research Paper

Authors

Department of Mathematics, Saurashtra University, Rajkot, India.

10.29252/asta.4.1.57

Abstract

 
‎The rings considered in this article are commutative with identity which admit at least two maximal ideals‎.  ‎This article is inspired by the work done on the comaximal ideal graph of a commutative ring‎. ‎Let R be a ring‎.  ‎We associate an undirected graph to R denoted by \mathcal{G}(R)‎,  ‎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 I1‎, ‎I2are adjacent in \mathcal{G}(R) if and only if I1∩ I2 = I1I2‎.  ‎The aim of this article is to study the interplay between the graph-theoretic properties of \mathcal{G}(R) and the ring-theoretic properties of R.

Keywords


[1] D.D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra. 159 (2) (1993), 500-514.
[2] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, (1969).
[3] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, (2000).
[4] I. Beck, Coloring of commutative rings, J. Algebra. 116 (1988), 208-226.
[5] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (4) (2011), 727-739.
[6] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (4) (2011), 741-753.
[7] R. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York, (1972).
[8] C. Gottlieb, Strongly prime ideals and strongly zero-dimensional rings, J. Algebra Appl. 16 (10) (2017) Article ID. 1750191 (9 pages).
[9] M.I. Jinnah and S.C. Mathew, When is the comaximal graph split?, Comm. Algebra. 40 (7) (2012), 2400-2404.
[10] H.R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra. 319 (2008), 1801-1808.
[11] S.M. Moconja and Z.Z. Petrovic, On the structure of comaximal graphs of commutative rings with identity, Bull. Aust. Math. Soc. 83 (2011), 11-21.
[12] K. Samei, On the comaximal graph of a commutative ring, Canad. Math. Bull. 57 (2) (2014), 413-423.
[13] P.K. Sharma and S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra. 176 (1995), 124-127.
[14] H.J. Wang, Graphs associated to co-maximal ideals of commutative rings, J. Algebra. 320 (7) (2008), 2917-2933.
[15] M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra Appl. 11 (6) (2012), Article ID. 1250114 (14 pages).