/sci/ - Science & Math

If [math]P[/math] is a projective generator in an abelian category [math]\textbf{A}[/math] and [math]R = Hom_{ \textbf{A}} (P, P)[/math], I've made a list of questions I will look into:
>if [math]R[/math] is a localized/commutative/noetherian/semilocal ring, what can be said about [math]\textbf{A}[/math]?
>can i import a notion analogous to Morita equivalences into abelian categories using the endomorphism rings like [math]R[/math]?
>what if [math]R-\textbf{mod}[/math] or the subcategory equivalent to [math]\textbf{A}[/math] is a localized category?
These should keep me busy for a while.