Generalization of prime submodules

In this thesis, prime ideals and prime submodules are examined within a theoretical framework, and their properties as well as generalized forms are studied in detail. In the first part of the study, the fundamental definitions, preliminary concepts, and existing approaches found in the relevant literature are systematically presented to establish a conceptual foundation; subsequently, certain extensions of the module concept are explored. In the second part, proof techniques based on theorems concerning structures such as FI-prime, S-prime, strongly prime, and completely prime submodules are analyzed. These techniques are employed in the findings section of the thesis. The original contribution of this thesis is the introduction of a new class of prime ideals and prime submodules that has not been previously defined in the literature. In this context, structures named A-prime ideal, A-domain, A-semiprime ideal, A-radical, and A-prime submodule are introduced; the conditions for their existence are identified, their fundamental structural properties are established, and these are supported by appropriate mathematical examples. Furthermore, the relationships between A-prime structures and classical prime structures are thoroughly analyzed, and similarities and differences between the two concepts are systematically compared. The findings of this study not only contribute to module and ideal theory but also provide a theoretical basis for future advanced research in this field.