Pw-closed Fuzzy Submodules and Pw-extending Fuzzy Modules

Abstract

This paper develops a structural theory of pseudo w-closed (pw-closed) fuzzy submodules and the associated class of pw-extending fuzzy modules. The proposed condition extends the w-closed framework and is distinguished from related notions by inclusion results, examples, counterexamples, and correspondence theorems. Under condition (*), pw-closedness of a fuzzy submodule is equivalent to pw-closedness of its associated classical submodule, establishing a direct link between the fuzzy and classical settings. Further results concern direct sums, complete essentiality, full semi-primality, y-closedness, level constructions, and preservation under suitable surjective F-homomorphisms. The pw-extending class is then studied through its relations with w-extending and extending fuzzy modules, its inheritance under direct summands, and its connections with FI-extending and CLS-(F-)modules. Examples and counterexamples clarify the scope of the principal implications and distinguish the proposed class from neighboring notions. The results provide a structural basis for further investigations of generalized closedness and extending properties in fuzzy module theory.