Vague Modules

Vague sets can distinguish the positive evidence from the negative evidence. But they only possess some simple operations such as intersection, union and complement of sets. Fusing Vague sets and fuzzy groups, Vague groups were proposed in 2005. Vague groups possess one type of the binary operations between elements. Some structures of Vague sets had been obtained. It is possible to further improve Vague groups by means of more operations between elements and the action of other objects on the Vague groups. In order to solve the problem, this paper presents the Vague rings and the Vague modules. The latter is one Vague group upon which the one Vague ring acts. In an Vague module, there are four types of operations: two types of operations in Vague ring, one type of operation in Vague group, and the operation between Vague ring and Vague group. Thus, this paper gives the six theorems: 1. the sufficient condition of the homomorphism commutative diagram of Vague modules; 2. the Vague free module are the Vague project modules; 3 some new internal and external structures of the project modules; 4. the uniqueness of project modules; 5. the Vague module is the direct sum of its Vague-submodules under the ascending chain condition; and 6. Vague Schur theorem. They add the new properties of internal and external algebraic structures to Vague groups. They have not been seen in literature about Vague groups.