AbstractBaer’s Criterion for Injectivity is a useful tool of the theory of modules.
Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith
[Algebra. II. Ring Theory,
Springer, Berlin, 1976]).
Recently, it has turned out that there are two classes of non-right perfect rings:
(1) those for which DBC fails in ZFC, and
(2) those for which DBC is independent of ZFC.
First examples of rings in the latter class were constructed in
[J. Trlifaj,
Faith’s problem on R-projectivity is undecidable,
Proc. Amer. Math. Soc. 147 2019, 2, 497–504];
here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.