Endo-Greatly Bounded Modules

<ns3:p>Background In the present paper, assuming every module H is unitary and every ring S is commutative with identity, we address the concepts of bounded modules and Endo-R.B.modules. We provide some examples, corollaries, remarks, and properties our new concept. An Endo-R.B S-module is considered to be more powerful than bounded module and Endo-R.B module that will present both of them in some details in the introduction. Motivated by these notions, we introduce and discuss a new class of modules called the Endo-Greatly Bounded module (abbreviated as Endo-G.B. module), a concept which has not been previously reported in the literature. Methods We formally define the Endo-G.B. module and provide examples, corollaries, and properties to illustrate this new concept. The study employs scalar modules and fully invariant submodules as crucial tools to establish connections. It must be emphasized that scalar modules and prime played a major role in achieving new results in this paper. We specifically investigate the important relationships between an Endo-G.B module and prime module as well as their generalizations. Results We get and prove that every Endo-G.B. module is Bounded, where as the converse is not generally true. Furthermore, we establish new results connecting our new class of S-modules to cyclic bounded modules, projective modules, multiplication modules, and finitely annihilated modules. We also derive necessary and sufficient conditions for an Endo-G.B. module to exhibit specific flexible properties. Conclusions In this paper, we presented a new class of S- modules called Endo-greatly bounded modules, these findings highlight the depth and flexibility of Endo-G.B. modules with in modern algebraic frameworks. This study opens new directions for further research in module theory, particularly in exploring the interaction between bounded ness and endomorphism structures, and may motivate future algebraic investigations in commutative algebra.</ns3:p>