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Let R be a commutative ring with unity and an R-submodule N is called semimaximal if and only if  the sufficient conditions of F-submodules to be semimaximal .Also the concepts of (simple , semisimple) F- submodules and quotient F- modules are  introduced and given some  properties .

In this paper, we introduce and study the essential and closed fuzzy submodules of a fuzzy module X as a generalization of the notions of essential and closed submodules. We prove many basic properties of both concepts.

This research analyses the correlation that exists between fuzzy T-ABSO and fuzzy socle. To answer this question, we define a new concept called, fuzzy socle T-ABSO submodules. Results showed many properties that satisfied the new notion. In addition, we applied some algebra operations and found the relationship between fuzzy T-ABSO and fuzzy socle -T-ABSO. We also investigated the structure of fuzzy socle T-ABSO sub- modules of a module under fuzzy direct sum, homomorphic image and intersection. Moreover, the relation between fuzzy socle T-ABSO sub-module and other types of fuzzy sub-module was presented. The implications of this study could be used to define new concepts that depend on fuzzy socle T-ABSO concepts

In this paper, we introduce and study the notions of fuzzy quotient module, fuzzy (simple, semisimple) module and fuzzy maximal submodule. Also, we give many basic properties about these notions.