Categories Of Modules Over Endomorphism Rings (memoirs Of The American Mathematical Society)

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, $\mathrm{Hom}(A, \cdot)$. The main result is a factorization $\mathrm{q}_{A}\mathrm{t}_{A}$ of the left adjoint $\mathrm{T}_{A}$ of $\mathrm{Hom}(A, \cdot)$, where $\mathrm{t}_{A}$ is a category equivalence and $\mathrm{ q}_{A}$ is a forgetful functor. Applications include a characterization of the finitely generated submodules of the right $E$-modules $\mathrm{Hom}(A,G)$, 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. It is the goal of the memoir to develop a functorial transfer of properties between [italic capital]A and [script capital]M[subscript italic capital]E, the category of modules over [italic capital]E, that is more sensitive than the traditional starting point, Hom([italic capital]A, ·). This memoir should be accessible to anyone who has a working knowledge of rings, modules, functors, and categories equivalent to that gained by reading Anderson and Fuller's text "Rings and Categories of Modules."