Exploring some modules through inclusion hypergraphs

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Baolsar, Iran.

Abstract

Recent studies have shown that hypergraphs are useful in solving real-life problems. Hypergraphs have been successfully applied in various fields. Inspired by the importance, we introduce a new hypergraph assigned to a given module. In particular, vertices of this hypergraph (which we call inclusion hypergraph, denoted by $InH_R(M)$) are all nontrivial submodules of a module $M$ and a subset $E$ of the vertices is a hyperedge in case each two elements of $E$ are comparable by inclusion and $E$ is maximal with respect to this condition. We prove that the inclusion hypergraph of an $R$-module $M$ is disconnected if and only if $M$ can be written as a direct sum of its each two nontrivial submodules. The diameter of $InH_R(M)$ is shown to be at most $3$.

Keywords

References

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Algebraic Structures and Their Applications

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Available Online from 01 November 2025

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