Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings (Q2122214)

A celebrate theorem of Baer states that an \(R\)-module \(Q\) is injective iff it is \(R\)-injective, i.e., for every monomorphism \(\iota:I\to R\) the morphism \(\mathrm{Hom}_R(\iota,Q)\) is an epimorphism. A similar result cannot be stated for projective modules over general rings [\textit{J. Šaroch} and \textit{J. Trlifaj}, Rend. Semin. Mat. Univ. Padova 144, 217--238 (2020; Zbl 1477.16003)]. On the other side, in [\textit{H. Alhilali} et al., J. Algebra 484, 198--206 (2017; Zbl 1384.16001)] the authors prove that the projectivity can be tested by using a set of epimorphisms only if the ring is perfect. Moreover, the present author proved that there are non-perfect rings with the property that such tests are possible if we impose some cardinal restrictions [Forum Math. 32, No. 3, 663--672 (2020; Zbl 1457.16002)] and that the coincidence of \(R\)-projectivity and projectivity for some commutative non-noetherian rings is consistent with ZFC+GCH, [\textit{J. Trlifaj}, Proc. Am. Math. Soc. 147, No. 2, 497--504 (2019; Zbl 1423.16003)]. In the present paper, the author studies connections between various projective properties for infinite Loewy length extensions of simple Artinian rings by using set-theoretic tools. In particular, he proves in Theorem 3.2 that if we assume some reasonable set theoretic hypotheses then all weakly \(R\)-projective modules over a small ring \(R\), and hence all \(R\)-projective modules, are projective.