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Faith’s problem on \(R\)-projectivity is undecidable. (English) Zbl 1423.16003

Baer criterion says that a right \(R\)-module \(M\) is injective if and only if \(M\) is \(R\)-injective, that is, each \(R\)-module homomorphism from any right ideal of \(R\) into \(M\) extends to \(R\). As the dual version of \(R\)-injectivity, a right \(R\)-module is called \(R\)-projective if each \(R\)-module homomorphism from \(M\) into \(R/I\), where \(I\) is any right ideal of \(R\), factors through the canonical projection \(R\rightarrow R/I\).
In view of Baer criterion, dual Baer criterion can be established as follows: a right \(R\)-module \(M\) is projective if and only if \(M\) is \(R\)-projective. However, there exists a module for which dual Baer criterion does not hold true. In other words, there exists an \(R\)-projective module which is not projective (note that if \(M\) is projective, then \(M\) is \(R\)-projective). A ring \(R\) such that dual Baer criterion holds true (i.e., for any right \(R\)-module \(M\), \(M\) is projective if and only if \(M\) is \(R\)-projective) is called right testing.
In [Algebra. Vol. II: Ring theory. Springer, Berlin (1976; Zbl 0335.16002)], C. Faith asked for what rings \(R\) does dual Baer Criterion hold true in the category \(\text{Mod}_R\) of right \(R\)-modules, that is, when does \(R\)-projectivity imply projectivity of all right \(R\)-modules? In [Relative injectivity and projectivity. State College: Pennsylvania State University Park (Ph.D. Thesis) (1964)], F. Sandomierski, proved that all right perfect rings are right testing. Recently, in [J. Algebra 484, 198–206 (2017; Zbl 1384.16001)], H. Alhilali et al. have shown for a number of nonright perfect rings that they are not right testing.
Let \(K\) be a field of cardinality \(\leq 2^{\aleph_0}\) and \(R\) the subalgebra of \(K^{\omega}\) consisting of all eventually costant sequences in \(K^{\omega}\). Then obviously \(R\) is not right perfect. Assume Gödel’s Axiom of Costructibility. Then the author shows an interesting result, which is, \(R\) is right testing (by using Jensen-functions) (i.e., every \(R\)-projective right \(R\)-module is projective). Thereby, when \(K\) is a field of cardinality \(\leq 2^{\aleph_0}\), the statement “\(R\) is right testing” is independent of ZFC+GCH. So the answer to the Faith’s question above is undecidable in ZFC+GCH.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras
03E35 Consistency and independence results
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
03E45 Inner models, including constructibility, ordinal definability, and core models
18G05 Projectives and injectives (category-theoretic aspects)
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References:

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