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Study on the New Graph Constructed by a Commutative Ring | ||
Theory of Approximation and Applications | ||
مقاله 1، دوره 12، شماره 1، مرداد 2018، صفحه 1-9 اصل مقاله (292.17 K) | ||
نوع مقاله: Research Articles | ||
نویسنده | ||
Zahra Yarahmadi* | ||
Department of Mathematics, Faculty of Science, Khorramabad Branch, Islamic Azad University, Khorramabad, I. R. Iran | ||
چکیده | ||
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. | ||
کلیدواژهها | ||
Connected graph؛ diameter؛ n-partite graph؛ Commutative ring؛ Noetherian؛ Artinian ring | ||
مراجع | ||
[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. | ||
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