We study the existence of covers and envelopes by some special functors on the category of
finitely presented modules. As an application, we characterize some important rings using
these functors. We also investigate homological properties of some functors on the stable
module category. The relationship between phantoms and Ext-phantoms is obtained. It is
shown that every left R-module M has an Ext-phantom preenvelope f: M→ N with coker (f)
pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab),(mod-R, Ab) is …