Study on the New Graph Constructed by a Commutative Ring

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)\۴(R). Also we study on the structure of ringR by some properties of vertices of subgraph G(R)\۴(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.

Keywords:

Connected graph , diameter , n-partite graph , Commutative ring , Noetherian , Artinian ring