FURTHER STUDIES OF THE PERPENDICULAR GRAPHS OF MODULES

In this paper we continue our study of perpendicular graph of modules, that was introduced in [7]. Let R be a ring and M be an R-module. Two modules A and B are called orthogonal, written A ⊥ B, if they do not have non-zero isomorphic submodules. We associate a graph Γ⊥(M) to M with vertices M⊥ = {(0) ̸= A ≤ M | ∃(0) ̸= B ≤ M such that A ⊥ B}, and for distinct A,B ∈M⊥, the vertices A and B are adjacent if and only if A ⊥ B. The main object of this article is to study the interplay of module-theoretic properties of M with graph-theoretic properties of Γ⊥(M). We study the clique number and chromatic number of Γ⊥(M). We prove that if ω(Γ⊥(M)) ∞ and M has a simple submodule, then χ(Γ⊥(M)) ∞. Among other results, it is shown that for a semi-simple module M, ω(Γ⊥(RM)) = χ(Γ⊥(RM)).