# Study on the New Graph Constructed by a Commutative Ring

Document Type: Research Articles

Author

Abstract

Let R be a commutative ring and G(R) be a graph with vertices as proper and
non-trivial ideals of R. Two distinct vertices I and J are said to be adjacent
if and only if I + J = R. In this paper we study a graph constructed from
a subgraph G(R)\Δ(R) of G(R) which consists of all ideals I of R such that
I Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper we
study about the relation between the number of maximal ideal of R and the
number of partite of graph G(R)\4(R). Also we study on the structure of ring
R by some properties of vertices of subgraph G(R)\4(R). In another section,
it is shown that under some conditions on the G(R), the ring R is Noetherian
or Artinian. Finally we characterize clean rings and then study on diameter
of this constructed graph.

Keywords

### References

[1]D.D. Anderson, V.P. Camillo, Commutative rings whose elements are a
sum of a unit and idempotent, J. Comm. Algebra 30 (2002), 3327-3336.

[2]I. Beck, Coloring of commutative rings, J. Algebra 116 (1998) 208-226.

[3]G. Chartrand, O.R. Oellermann, Applied and Algorithmic Graph Theory,
McGraw-Hill, Inc., New York, 1993.

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

[5]C. Mih aila, Comaximal Ideal Graphs of Commutative Rings, Wellesley
College, Honors Thesis Collection, 2012.

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

[7]R. Y. Sharp, Steps in commutative algebra, Londen Mathematical Society
Student Texts51, Cambridge University Press, Cambridge, 2000.

[8]D.B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle
River, NJ, 1996.