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 ?



For which rings does a projectivization of modules exist?
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.


