Categories of Modules over Endomorphism Rings

The goal of this work is to develop a functorial transfer of properties between a module $A$ and the category ${\mathcal M}_{E}$ of right modules over its endomorphism ring, $E$, that is more sensitive than the traditional starting point, $\textnormal{Hom}(A, \cdot )$. The main result is a factorization $\textnormal{q}_{A}\textnormal{t}_{A}$ of the left adjoint $\textnormal{T}_{A}$ of $\textnormal{Hom}(A, \cdot )$, where $\textnormal{t}_{A}$ is a category equivalence and $\textnormal{ q}_{A}$ is a forgetful functor. Applications include a characterization of the finitely generated submodules of the right $E$-modules $\textnormal{Hom}(A,G)$, a connection between quasi-projective modules and flat modules, an extension of some recent work on endomorphism rings of $\Sigma$-quasi-projective modules, an extension of Fuller's Theorem, characterizations of several self-generating properties and injective properties, and a connection between $\Sigma$-self-generators and quasi-projective modules.