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Applying a hypergraph to determine the structure of some finite modules

Journal of Applied Mathematics and Computing, ISSN: 1865-2085, Vol: 69, Issue: 1, Page: 675-687
2023
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Article Description

Characterization of a finite module with specified number of nontrivial submodules is one of the most important issues for researchers in module theory. In this paper, we will try to characterize a module with three or four nontrivial submodules by defining a new hypergraph on that module. Suppose that K is a module over a ring R. We introduce IH(K) which we call intersection hypergraph of K. Any hyperedge in IH(K) , forms a complete subgraph of the complement of intersection graph of a module. A characterization of a finite module with exactly three nontrivial submodules via their associated hypergraphs are also presented. We provide a characterization of finite semisimple modules with exactly four nontrivial submodules in terms of their corresponded hypergraph. Some interesting examples are also included.

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