A Comparative study Between R-modules and Fuzzy R-modules (Sebha University Conference Proceedings, 2025)
A comparative study on the concepts and structures of classical modules in crisp set theory and fuzzy modules is presented in this article. We discuss modules, fuzzy modules, fuzzy submodules, fuzzy module homomorphisms, and fuzzy quotient modules. The paper focuses on differences in definitions and basic properties of crisp and fuzzy modules. We show that the extension from classic to fuzzy modules is a natural one and that many module-theoretic notions in the classical case can be naturally extended to hold in the fuzzy case; this may be viewed as an indication that the class of fuzzy modules generalizes that of classical modules. This extension underscores the flexibility of module theory in fuzzy environments, accommodating uncertainty and imprecision at least.