Uncovers connections between module-theoretic and graph-theoretic properties in modules, suggesting new relationships.
In this paper we continue our study of perpendicular graph of modules, that was introduced in {Hokkaido}. 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_(R M))=χ(Γ_(R M)).
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SHIRALI et al. (2023) studied this question.
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