Document Type : Research Note


Department of Mathematics, University of Yasouj, Yasouj, Iran.


‎In this paper we continue our study of perpendicular graph of modules‎, ‎that was introduced in \cite{Hokkaido}‎. ‎Let $R$ be a ring and $M$ be an $R$-module‎. ‎Two modules $A$ and‎
‎$B$ are called orthogonal‎, ‎written $A\perp B$‎, ‎if they do not have‎
‎non-zero isomorphic submodules‎. ‎We associate a graph $\Gamma_{\bot}(M)$ to $M$‎
‎with vertices‎
‎$\mathcal{M}_{\perp}=\{(0)\neq A\leq M\;|\; \exists (0)\neq B\leq M \; \mbox{such that}\; A\perp B\}$‎,
‎and for distinct $A,B\in‎
‎\mathcal{M}_{\perp}$‎, ‎the vertices $A$ and $B$ are adjacent if and only if‎
‎$A\perp B$‎. ‎The main object of this article is to study the‎
‎interplay of module-theoretic properties of $M$ with‎
‎graph-theoretic properties of $\Gamma_{\bot}(M)$‎. ‎We study the clique number and chromatic number of $\Gamma_{\bot}( M)$‎. ‎We prove that if $\omega(\Gamma_{\bot}( M)) < \infty $ and $M$ has a simple submodule‎, ‎then $\chi(\Gamma_{\bot}(M)) < \infty $‎. ‎Among other results‎, ‎it is shown that for a semi-simple module $M$‎, ‎$\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$‎.


1. S. Akbari, H. A. Tavallaee and S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl., 11 (2012), Article ID: 1250019.
2. A. Amini, B. Amini, E. Momtahan and M. H. Shirdareh Haghighi, On a graph of ideals, Acta Math. Hungar, 134 (2012), 369–384.
3. J. Dauns and Y. Zhou, Classes of modules, Chapman and Hall, 2006.
4. O. A. S. Karamzadeh and A. R. Sajedinejad, Atomic modules, Comm. Algebra, 29 (2001), 2757–2773.
5. B. Krön, End compactifications in non-locally-finite graphs, Math. Proc. Cambridge Philos. Soc., 131(3) (2001), 427–443.
6. A. Ç. Özcan, A. Harmanci and P. F. Smith, Duo modules, Glasg. Math. J., 48 (2006), 533–545.
7. M. Shirali, E. Momtahan and S. Safaeeyan, Perpendicular graph of modules, Hokkaido Math. J., 49 (2020), 463–479.
8. B. Stenström, Rings of quotients, Springer-Verlag, New York, 1975.