Let $R$ be a ring with unity. In this paper, we introduce a new graph associated with $R$ called the simple-intersection graph of $R$, denoted by $GS(R)$. The vertices of $GS(R)$ are the nonzero ideals of $R$, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of $R$, and the graph properties of $GS(R)$ such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of $GS(R)$, as well as give a formula to compute the clique and domination numbers of $GS(R)$.