Study on the New Graph Constructed by a Commutative Ring

Document Type: Research Articles


Department of Mathematics, Faculty of Science, Khorramabad Branch, Islamic Azad University, Khorramabad, I. R. Iran


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.


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