TITLE: For which rings does a projectivization of modules exist?
QUESTION [6 upvotes]: 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 ?

REPLY [6 votes]: A necessary condition is that, for any finitely generated R-module M, the Hom-object $Hom_R(M,R)$ is a finitely generated projective right R-module.  This is because if a left adjoint existed, this would be isomorphic to the set $Hom_R("Proj(M)",R)$, and this is a summand of a free right module.
Consider the case $R = \mathbb{Z}/p^2$, $M = \mathbb{Z}/p$, $P = R$.  Then $$Hom_{\mathbb{Z}/p^2}(\mathbb{Z}/p, \mathbb{Z}/p^2) \cong \mathbb{Z}/p$$
and so this ring admits no projectivization.
Conversely, if this condition is satisfied then we can define $D_R M = Hom_R(M,R)$, and $DM$ is always finitely generated projective with a natural double-duality map $M \to D_{R^{op}} D_R M$ which is an isomorphism for projective modules.  This provides the desired adjoint, because any map $M \to P$ produces a natural factorization
$$M \to DDM \to DDP \leftarrow^{\sim} P$$
which is the desired adjoint.