In this article, we introduce the fuzzy generator and fuzzy cogenerator and investigate some of their properties. Necessary and sufficient conditions for a fuzzy module to become a fuzzy generator and sufficient conditions for a fuzzy module to become a fuzzy cogenerator have also been established. We also discuss the behavior of fuzzy modules of finitely generated modules. Further, we introduce fuzzy trace and fuzzy reject and establish some specific results. Finally, we define a faithful fuzzy module and weave this notion with the concepts of fuzzy generator and fuzzy cogenerator.
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