Is it true that if $\operatorname{Ext}^{1}\_{A}(P,A/I)=0$ forall $I$ then $P$ is projective? 


Similar statements are true for flat and injective modules, but I'm beginning to suspect that projective modules cannot be characterized solely by ideals.



Is it true that if $\operatorname{Ext}^{1}_{A}(P,A/I)=0 $ for all $ I$ then $P$ is projective?
Answer is yes if $A$ is Noetherian and $P$ is finitely generated. Indeed, your 
condition
implies that $Ext^1(P, N)=0$ for any finitely generated $A$-module $N$, which implies that
$P$ is projective.


