Studying some semisimple modules via hypergraphs

Recent studies have shown that hypergraphs are useful in solving real-life problems. Hypergraphs have been successfully applied in various fields. Inspiring by the importance, we shall introduce a new hypergraph assigned to a given module. By the way, vertices of this hypergraph (which we call sum hypergraph) are all nontrivial submodules of a module $P$ and a subset $E$ of the vertices is a hyperedge in case the sum of each two elements of $E$ is equal to $P$ and $E$ is maximal with respect to this condition. Some general properties of such hypergraphs are discussed. Semisimple modules with length $2$ are characterized by their corresponding sum hypergraphs. It is shown that the sum hypergraph assigned to a finite module $P$ is connected if and only if $P$ is semisimple.