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

Herden, Daniel; Salazar Pedroza, Héctor Gabriel

Commentat. Math. Univ. Carol. 57, No. 1, 7-20 (2016).

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