<![CDATA[This study analyzes prime submodules and primality properties in module spaces, fundamental topics in abstract algebra. The primary objective is to determine conditions for prime submodule formation in commutative rings and to examine their relationship with nearly prime submodules. The approach employed involves the decomposition of modules in principal ideal domains, with particular attention given to the concepts of annihilators, submodule orders, and the characteristics of torsion-free quotient modules. The findings indicate that nearly prime submodules can only be regarded as prime submodules in free modules. Furthermore, the module decomposition approach proves effective for gaining a comprehensive understanding of module structures. The concepts of annihilators and submodule orders provide profound insights into the relationships between module elements and ring elements. This study offers significant theoretical contributions to abstract algebra and establishes a foundation for further developments, particularly in applications related to number theory and cryptography.]]>
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