Planar, outerplanar and ring graph of the intersection graph

Let R be a commutative ring and M be an R-module. The M-intersection graph of ideals of R, denoted by G_M(R) is a graph with the vertex set I(R)^*, and two distinct vertices I and J are adjacent if and only if IM\cap JM\neq ۰. In this paper, we study G_{R/J}(R/I), where I and J are ideals of R and I\subseteq J. We characterize all ideals I and J for which G_{R/J}(R/I) is planar, outerplanar or ring graph.

Keywords:

Intersection graph , Outerplanarity , Planarity , Ring graph