A Kind of Intersection Graphs on Ideals of a Ring

Problem statement: Let R be a ring. The graph G(R) is the graph whose vertices are nontrivial ideal of R and in which two vertices u, v are joined by an edge, if and only if u ∩ v #{0}. Approach: In this study we study some properties of G(R). Results: We obtain conditions of R such that G(R) is a path and determine the graph G(R) in which it is a tree. Conclusion: We conclude that ideals of R have degree one.