The Properties Of z-Essential (z-Closed) Fuzzy Submodules
Abstract
The theory of fuzzy submodules, rooted in the broader domain of fuzzy set theory, provides a nuanced framework for studying algebraic structures under uncertainty. Traditional studies on essential and closed fuzzy submodules have offered important foundations, yet they fall short in addressing deeper generalizations required for more complex fuzzy module interactions. While essential and closed fuzzy submodules are well-established, their extensions namely, -essential and -closed fuzzy submodules have received limited attention, particularly concerning their characterizations and behavior under module homomorphisms. This study aims to define and investigate the properties of -essential and -closed fuzzy submodules within -fuzzy modules, exploring their structural implications and interrelations with classical fuzzy submodule concepts. The research establishes that -essential fuzzy submodules generalize essential ones and introduces conditions under which a -essential submodule becomes essential. Similarly, it proves that every -closed submodule is closed, though not conversely, and provides conditions for closed submodules to be -closed. Moreover, the behavior of -closed submodules under homomorphisms and their preservation through module operations is rigorously examined. This study pioneers the formalization of -essential and -closed fuzzy submodules, offering novel characterizations and multiple propositions that extend classical fuzzy module theory. These results enhance the understanding of structural properties in fuzzy algebra and provide a basis for further exploration in module theory under fuzzified environments, with potential applications in computational algebra and systems modeling where partial membership and graded structures are relevant.
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