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On Whitehead modules. (English) Zbl 0743.16004
The authors prove consistency results with ZFC+GCH, related to Whitehead and Baer modules. Their use of forcing is essential in proofs of the following main results. For a wide class of rings \(R\) it is consistent that, for every \(R\)-module \(K\) there is a non-projective module \(M\) with \(Ext^ 1_ R(M,K)=0\); consequently, there are non-projective Whitehead \(R\)-modules. The generalization is the consistency of the statement that, for some rings \(R\), there exist Whitehead \(R\)-modules which are not the union of a continuous chain of submodules with all the quotients being small Whitehead \(R\)-modules. It is undecidable in ZFC+GCH whether there is a test module for being a Baer module.

MSC:
16D40 Free, projective, and flat modules and ideals in associative algebras
16B70 Applications of logic in associative algebras
03C25 Model-theoretic forcing
03C60 Model-theoretic algebra
03E35 Consistency and independence results
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