## 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

### Keywords:

prediction principles; almost free modules; dual modules
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