Support Based Essential and Core Based Superfluous Fuzzy Modules

In this paper, we introduce and explore several novel concepts within the framework of fuzzy module theory. First, we define the notion of a support based essential fuzzy module, establishing its foundational properties. We then investigate the essentiality of alpha cuts and the quotient fuzzy modules arising from support based essential fuzzy modules. Additionally, we demonstrate that under specific conditions, the product of a fuzzy ideal and a fuzzy module results in a support based essential fuzzy module. Further, we define the concept of a support based essential fuzzy monomorphism and provide a detailed characterization. We also introduce the fuzzy injective hull and prove that the direct sum of the fuzzy injective hulls of a class of fuzzy modules coincides with the fuzzy injective hull of the direct sum of the same class. Finally, we define the core based superfluous fuzzy module as a dual concept to the support based essential fuzzy module and establish corresponding dual results.