Study on the New Graph Constructed by a Commutative Ring
محورهای موضوعی : Applied Mathematics
زهرا یاراحمدی
1
(Department of Mathematics, Faculty of Science, Khorramabad Branch,
Islamic Azad University, Khorramabad, I. R. Iran)
کلید واژه: diameter, Connected graph, n-partite graph, Commutative ring, Noetherian, Artinian ring,
چکیده مقاله :
Let R be a commutative ring and G(R) be a graph with vertices as proper andnon-trivial ideals of R. Two distinct vertices I and J are said to be adjacentif and only if I + J = R. In this paper we study a graph constructed froma subgraph G(R)\Δ(R) of G(R) which consists of all ideals I of R such thatI Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper westudy about the relation between the number of maximal ideal of R and thenumber of partite of graph G(R)\4(R). Also we study on the structure of ringR 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 Noetherianor Artinian. Finally we characterize clean rings and then study on diameterof this constructed graph.
[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 aila, 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.
