A graph which recognizes idempotents of a commutative ring
  In this paper we introduce and study a graph on the set of ideals of a commutative ring $R$. The vertices of this graph are non-trivial ideals of $R$ and two distinct ideals $I$ and $J$ are adjacent if and only $IJ=I\cap J$. We obtain some properties of this graph and study its relation to the structure of $R$. 
Hamid Reza Dorbidi and Saeid Alikhani
math.CO