Is it true that if $\mbox{Ext}^{1}_{A}(P,A/I)=0 \ \forall I$$\operatorname{Ext}^{1}_{A}(P,A/I)=0$ forall $I$ then P$P$ is projective?
Similar statements are true for flat and injective modules, but I'm beginning to suspect that projective modules cannot be characterized soleysolely by ideals.