Abstract 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 [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.

中文翻译：

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非完美戒指的双重贝尔标准

摘要 Baer's Criterion for Injectivity是模理论的一个有用的工具。众所周知，它的对偶版本 (DBC) 适用于所有正确的完美环，但它对非正确完美环的有效性是一个复杂的问题（首先由 C. Faith 提出 [Algebra. II. Ring Theory, Springer, Berlin, 1976] ]）。最近，已经证明有两类非右完美环：（1）DBC 在 ZFC 中失败的那些，以及（2）DBC 独立于 ZFC 的那些。后一类中环的第一个例子在[J. Trlifaj，信仰关于 R 投影的问题是不可判定的，Proc。阿米尔。数学。社会。147 2019, 2, 497-504]；在这里，我们展示了这个类包含所有具有原始因子 artinian 的小半亚丁型冯诺依曼规则环。