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Separable \(\aleph_k\)-free modules with almost trivial dual. (English) Zbl 1374.13011

Summary: An \(R\)-module \(M\) has an almost trivial dual if there are no epimorphisms from \(M\) to the free \(R\)-module of countable infinite rank \(R^{(\omega)}\). For every natural number \(k>1\), we construct arbitrarily large separable \(\aleph_k\)-free \(R\)-modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

MSC:

13B10 Morphisms of commutative rings
13B35 Completion of commutative rings
13C13 Other special types of modules and ideals in commutative rings
13J10 Complete rings, completion
13L05 Applications of logic to commutative algebra
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