ON THE CO-MAXIMAL GRAPH OF MODULES

Let R be a ring and M be a left R module. Let Rm = { rm : r ∈ R } be a left R submodule of M. Suppose that ⌈(M) be a graph with vertex set V = M and edge set E ={e = ab : Ra+Rb = M }. Let ⌈1(M) be the vertices m in ⌈(M) such that Rm = M. Let ⌈2(M) = ⌈(M) \ ⌈1(M) and ⌈3(M) = ⌈2(M) \ J(M). We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph ⌈3(M). In addition, it is shown that for two nitely generated R modules M and N, if R is semisimple, then ⌈(M) ≅ ⌈(N) if and only if M ≅ N. Finally, the graph associated to quotient module M/J(M) is studied.