Investigating some concepts related with Z – essential submodules

Key Points

• Three new concepts have been introduced in module theory, contributing to the understanding of module structures.
• The concepts discussed include z - essential small submodules and ze - radical of a module, enhancing theoretical frameworks.
• The paper explores properties related to ze - socle of a module, providing new insights into module analysis.
• Implications of these concepts may influence future research directions in abstract algebra.