Intersection graph of a module

Let V be a left R-module where R is a (not necessarily commutative) ring with unit. The intersection graph {G}(V) of proper R-submodules of V is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper R-submodules of V, and there is an edge between two distinct vertices U and W if and only if U ∩ W ≠ 0. We study these graphs to relate the combinatorial properties of {G}(V) to the algebraic properties of the R-module V. We study connectedness, domination, finiteness, coloring, and planarity for {G} (V). For instance, we find the domination number of {G} (V). We also find the chromatic number of {G}(V) in some cases. Furthermore, we study cycles in {G}(V), and complete subgraphs in {G} (V) determining the structure of V for which {G}(V) is planar.