On Pr-Essential and Pr-Closed Fuzzy Submodules Characterizations Via Level Submodules

Hassan K Marhon

doi:10.35799/jis.v26i2.69295

Authors

Hassan K Marhon Ministry of Education, General Directorate of Education of Baghdad Al-Rusafa1, Iraq https://orcid.org/0009-0000-1641-5137

DOI:

https://doi.org/10.35799/jis.v26i2.69295

Keywords:

Pure fuzzy submodule, Pure essential fuzzy submodule, Pure closed fuzzy submodule, Essential fuzzy submodule, Fuzzy module theory

Abstract

This paper investigates pure-essential and pure-closed fuzzy submodules in this setting of fuzzy module theory. The concepts are developed by incorporating purity into the classical settings of essentiality and closedness for fuzzy submodules. Several equivalent characterizations are established, and structural properties of Pr-essential and Pr-closed fuzzy submodules are investigated, particularly through their corresponding level submodules. The obtained results clarify the relationships among pure fuzzy submodules, essential fuzzy submodules, closed fuzzy submodules, and their pure-related counterparts. Examples and counterexamples are provided to illustrate the developed concepts, distinguish the resulting properties from related fuzzy notions, and demonstrate that some converses do not hold in general. These findings extend existing results in fuzzy module theory and provide a level-submodule framework for analyzing purity-related conditions in fuzzy algebra. The study also indicates directions for further investigation of the interaction between essentiality, purity, and closedness in fuzzy modules.

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