Let $R$ be a ring. Consider the inclusion functor from the category of finitely generated, projective $R$-modules to the category of all finitely generated $R$-modules. For which rings does it have a left adjoint.
For example for any principal ideal domain $R$, we have the structure theorem of f.g. $R$-modules. The upper left adjoint is given by $M\mapsto M/torsion$.
In general there is no reason, why such a construction should exist. I tried to check the assumptions for Freyds adjoint functor theorem. But it seems quite hard to check whether, they are satisfied for a given ring $R$.
So does anyone know a counterexample, where such a adjoint doesn't exist or a weaker property than PID, that implies the existence ?