The Theory of T-Bipolar Soft Modules

Algebraic structures in mathematics and applications reveal an inherent duality or bipolarity that is seen in elements that have both positive and negative behaviors or properties. Integrating the already-known module theory concept with the capability of T-bipolar soft sets to handle such bipolar information can lead to discoveries and instruments that can be used to research and operate these algebraic structures. This integration of module theory and T-bipolar soft sets may result in new forms of analysis and methods of analysis. These new approaches could greatly improve the possibilities of studying, analyzing, and solving problems related to algebraic structures that are inherently bipolar. Thus, in this manuscript, we investigate the notion of a T-bipolar soft module. We also deduce the T-bipolar soft submodule, maximal T-bipolar soft submodule, and their associated results. Further, we investigate T-bipolar soft homomorphism, and T-bipolar soft isomorphism along with important results such as the First Isomorphism Theorem, Second Isomorphism Theorem, and Third Isomorphism Theorem in the structure of T-bipolar soft module. Additionally, we diagnose T-bipolar soft exactness and associated results.