Abstract
In Faith [Grundlehren der Mathematischen Wissenschaften. 191 (1976)], Faith asked for what rings R does the Dual Baer Criterion hold in Mod-R, that is, when does R-projectivity imply projectivity for all right R-modules? Such rings R were called right testing. Sandomierski proved that all right perfect rings are right testing.
Puninski et al. [J. Algeb. 484 (2017) pp. 198-206] have recently shown for a number of nonright perfect rings that they are not right testing, and noticed that [Trans.
Amer. Math.
Soc. 348 (1996) pp. 1521-1554] proved consistency with ZFC of the statement 'each right testing ring is right perfect' (the proof used Shelah's uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions).
Thus the answer to the Faith's question above is undecidable in ZFC. We also provide examples of nonright perfect rings such that the Dual Baer Criterion holds for all small modules (where small means countably generated, or <= 2(N0)-presented of projective dimension <= 1).