Perpendicular graph of modules

Let R be a ring and M be an R-module. Two modules A and B are called orthogonal, written A perpendicular to B, if they do not have non-zero isomorphic submodules. We associate a graph Gamma(perpendicular to)(M) to M with vertices M-perpendicular to = {(0) not equal A <= M vertical bar there exists B not equal (0) such that A perpendicular to B}, and for distinct A, B is an element of M-perpendicular to, the vertices A and B are adjacent if and only if A perpendicular to B. The main object of this article is to study the interplay of module theoretic properties of M with graph-theoretic properties of Gamma(perpendicular to)(M). An algorithm is given to generate perpendicular graphs of Z(n).