FURTHER STUDIES OF THE PERPENDICULAR GRAPHS OF MODULES | ||
| Journal of Algebraic Systems | ||
| دوره 12، شماره 2، فروردین 2025، صفحه 391-401 اصل مقاله (216.8 K) | ||
| نوع مقاله: Research Note | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2023.11606.1587 | ||
| نویسندگان | ||
| Maryam Shirali* ؛ Saeid Safaeeyan | ||
| Department of Mathematics, University of Yasouj, Yasouj, Iran. | ||
| چکیده | ||
| 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 $M$, $\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$. | ||
| کلیدواژهها | ||
| chromatic number؛ clique number؛ finite graph؛ atomic module؛ semi-simple module | ||
| مراجع | ||
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