Baer’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
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.
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