Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct vertices are connected if their product is zero. Lets call this graph, $G_R$.
Suppose that the chromatic number of $G_R$ is finite. I am interested in minimal prime ideals of $R$. My guess is that for any minimal prime ideal $P$ the ring $R_P$ is either a field or finite. Any suggestion or references would be helpful.
