Ebook: 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.