Let $R$ be a commutative ring. If necessary, assume that $R$ has any convenient properties you like.

- Is there some $R$-module $Q$ such that an $R$-module $P$ is projective if and only if $\hom_R(P,Q) \to \hom_R(P,Q')$ is surjective for all quotients $Q'$ of $Q$?
- If yes, can $Q$ be chosen as a cogenerator?
- What happens when we restrict to finitely generated $P$?

Neither $Q=R$ nor $Q=\hom_{\mathbb{Z}}(R,\mathbb{Q}/\mathbb{Z})$ work. This question is a follow-up of math.SE/325495.

class$\mathcal{C}$ of $R$-modules such that it is sufficient to check the surjectivity for every quotient $Q \rightarrow Q'$ with $Q$ in $\mathcal{C}$: namely, the class of injective modules. Or more generally, any class such that any module can be embedded into some object of this class. If under non-trivial circumstances this class can be chosen to be smaller, maybe to contain only one object, I don't know. $\endgroup$ – Torsten Schoeneberg Mar 12 '13 at 11:01Lectures on Modules and Rings, p. 125. $\endgroup$ – Torsten Schoeneberg Mar 12 '13 at 17:59