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user9072
user9072

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

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.

projective module

Is it true that if $\mbox{Ext}^{1}_{A}(P,A/I)=0 \ \forall I$ 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 soley by ideals.

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

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.

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ashpool
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projective module

Is it true that if $\mbox{Ext}^{1}_{A}(P,A/I)=0 \ \forall I$ 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 soley by ideals.