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 ۲ 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.

کلیدواژه ها:

Hyperedge ، Hypergraph ، Semisimple module ، Simple submodule ، Small submodule