Planar, outerplanar and ring graph of the intersection graph

Let $m,n>1$ be two integers, and ${\mathbb{Z}}_n$ be a ${\mathbb{Z}}_m$-module. Let $I({\mathbb{Z}}_m{)}^{\ast }$ be the set of all non- zero proper ideals of ${\mathbb{Z}}_m$. The ${\mathbb{Z}}_n$-intersection graph of ${\mathbb{Z}}_m$, denoted by $G_n({\mathbb{Z}}_m)$ is a graph with the vertex set $I({\mathbb{Z}}_m{)}^{\ast }$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I{\mathbb{Z}}_n\cap J{\mathbb{Z}}_n\mathrm{\ne }0$. In this paper, we determine the values of $m$ and $n$ for which $G_n({\mathbb{Z}}_m)$ is planar, outerplanar or ring graph.