Generating and cogenerating structures

A functor $T:\mathcal{A} \to \mathcal{B}$ acts faithfully on the right of a class of objects $\mathcal{A}'$ of $\mathcal{A}$ if it distinguishes morphisms out of objects of $\mathcal{A}'$ (that is, $A' \in \mathcal{A}',X \in \mathcal{A},f,g \in \mathcal{A}(A',X)$ and $f \ne g$ implies $T(f) \ne T(g))$. We define a full subcategory $\mathcal{R}\mathcal{F}(T)$ such that $T$ acts faithfully on the right of the objects of $\mathcal{R}\mathcal{F}(T)$. An object $U \in \mathcal{A}$ is a generator if ${H^U}:\mathcal{A} \to \mathcal{E}ns$ is faithful, and if ${H^U}$ is not faithful, we may still consider $\mathcal{R}\mathcal{F}({H^U})$. This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of $R$-modules. Relationships between these can be used in the homological classification of rings.