Over the last forty years, interest has arisen in the study of algebraic systems like Rings, Modules over Rings. As the concept ''prime ideal'' in Ring Theory is a generalization of prime number, the study of prime ideals occupied an important place in Algebra. Success of fuzzy logic in wide range applications inspired much interest in fuzzy logic among mathematicians. The researchers in mathematics were trying to introduce and study the concept of fuzzyness in different mathematical system under study. Dr Bhavanari Satyanarayana & his Research Group studied several important concepts like: Goldie/Spanning/Fuzzy Dimension, Prime Fuzzy Submodules in Modules over Associative Rings. The new Algebraic System ''Gamma Near-ring'' introduced by Dr Bhavanari, is now receiving good attention by the Mathemaical World. This Monograph covers a vital study of the concepts like: Prime Submodule, Prime Fuzzy Submodule, Fuzzy Coset of a Fuzzy Submodules in Modules. This monograph is certainly attracts the interest of all Ring Theorists. This Monograph is more valuable addition not only to the Institutional Libraries, but also to each personal library of each and every Algebraist of the world.