The Baer Criterion for Injectivity is a simple, but very useful tool for the classification of injective modules. In 1976, Carl Faith asked for what rings R does the Dual Baer Criterion hold, that is, when does R-projectivity imply projectivity for an arbitrary module M?

A positive answer had already been known for any ring R when the module M is finitely generated, and for all modules M when R is a perfect ring. Various ad hoc proofs showed that the answer is negative for many particular non-perfect rings. For example, for each commutative noetherian ring R of Krull dimension > 0 there is a countably generated R-projective module M which is not projective. However, for each infinite cardinal c, there is a non-perfect ring R_c such that the answer is positive for any R_c-module M of cardinality at most c.

In 2017, Puninski noticed that the proof of the consistency of the existence of non-projective Whitehead modules yields consistency of a negative answer to Faith’s question for each non-perfect ring R. Recently, it has turned out that the analogy with the Whitehead module setting goes even further: assuming the Weak Diamond, there is a class of non-perfect rings C consisting of transfinite extensions of full matrix rings over skew-fields, for which the answer is positive for any module M. The simplest ring in C is the Bergman ring of all eventually constant sequences of elements of a field.