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In order to test injectivity of a module $M$ it suffices to check if every linear map from an arbitrary ideal extends to the ring or not. Similarly in order to check the flatness of a module $M$ it suffices to check whether tensoring with it preserves injectivity of $0 \to I \to R$.

Is there an analogue of these statements for testing projectivity ?

May be if a full analogue is not there in the general case, can we have it if the ring satisfies some conditions like PID/Noetherian-ness ?

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    $\begingroup$ The analogue is for flatness. In a technical sense, the «correct» dual notion to injectivity is not projectivity but flatness. That is why, for example, there are injective envelopes and flat covers but not projective covers in general, and so on. $\endgroup$ – Mariano Suárez-Álvarez May 10 '14 at 20:02
  • $\begingroup$ @MarianoSuárez-Alvarez Can you state specifically the flat analogue? Thank you $\endgroup$ – rschwieb Jul 4 '14 at 13:19
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Let $N$ be a class of objects in the category of right $R$-modules ($R$ a ring with unity) such that any right $R$-module can be embedded in some module in $N$. (For example, $N$ may be taken the class of injective right $R$-modules.) In testing whether a right module $P$ is projective, it suffices to check that, for any $R$-epimorphism $g : B→C$ where $B$ and $C$ are right $R$-modules, any $R$-homomorphism $h : P→C$ could be lifted to a homomorphism $f : P→B$, i.e. $h=gof$.

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  • $\begingroup$ It seems that your criterion for testing projectivity is simply the definition. Did you mean to use the class $N$ somewhere? $\endgroup$ – zcn May 22 '14 at 21:08
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Trlifaj, Jan. "Faith's problem on R-projectivity is undecidable." arXiv preprint arXiv:1710.10465 (2017)

In the publication above, Trlifaj shows that the statement "each ring satisfying the dual Baer criterion is right perfect" is consistent with ZFC, and also its negation is consistent with ZFC, so in fact the problem is undecidable.

The "best" result then is that for right modules over right perfect rings, one can use the "dual Baer criterion."

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