A New Class of Endo-R.B Module and Its Relationship with Modules

This paper gives a definition of a new class of T-module and T-submodule called an Endo-Restricted Bounded module (submodule) written shortly by Endo-R.B. module (submodule) and present some different approaches to connect this class of module with other types of modules such as: compressible modules, monoform modules, critically compressible modules, retractable modules, and quasi-Dedekind modules. One of the main purpose of this work is to introduce a few new conditions and reveal some properties and corollaries. This paper considered to be another solution or answer for Zelmanowitz’s problem. In fact, an Endo-R.B. T-module plays an important role to this problem and we say under what condition compressible T-module would be a critically compressible T-module so we present a positive solution depend on an Endo-R.B module. The T-homomorphism of compressible and monoform modules join in a nice way with an endomorphism of a T-module Ω that we need it for the definition of an Endo-R.B module. Moreover, an Endo-R.B module gives us directly three different modules and these modules are bounded module, finitely annihilated module, and retractable module. Finally, polyform and fully polyform modules are involved in this article.