FURTHER STUDIES OF THE PERPENDICULAR GRAPHS OF MODULES

Document Type : Research Note

Authors

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

Abstract

‎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 $M$‎, ‎$\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$‎.

Keywords

References

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Journal of Algebraic Systems
Volume 12, Issue 2
January 2025
Pages 391-401

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