It is well-known that by Baer's Criterion, for a $R$-module $Q$, $\text{Hom}(-,Q)$ is exact (i.e. $Q$ is injective) if and only if resulting sequence of this functor act on exact sequence $$0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow0$$ is exact for any left ideal $I$.
Then is it true for analogous (or, in some sense, dual) statement that for a $R$-module $P$, $\text{Hom}(P,-)$ is exact (i.e. $P$ is projective) if and only if it is exact on special exact sequence shown above?
In fact my friend guess this is not true for some reason that I don't know. However, I can't find any counterexamples of this. So if there exist any counterexample please show me.
Also, If this is false, then is there exist some criterion for projectivity analogous to Baer's criterion? More precisely, is there exist certain class of exact sequence, depend on $R$, such that $\text{Hom}(P,-)$ is exact in that sequences implies projectivity?