$\aleph_n$-free Modules over complete discrete valuation domains with small dual

by Goebel and Shelah and Struengmann. [GbShSm:981]
Glasgow Math J, 2013
Let M^*Hom_R(M,R) be the dual module of M for any commutative ring R . In [GS] we applied a recent prediction principle from Shelah [Sh] to find aleph_n-free R-modules M with trivial dual M^*=0 for each natural number n . This can be achieved for a large class of rings R including all countable, principle ideal domains which are not fields. (Recall that an R-module M is kappa-free for some infinite cardinal kappa if all its submodules generated by < kappa elements are contained in a free R-submodule.) However, the result fails if R is uncountable, as can be seen from Kaplansky's [Ka] well-known splitting theorems for modules over the ring J_p of p-adic integers. Nevertheless we want to extend the main result from [GS] to complete discrete valuation domains (DVDs) R, in particular to p-adic modules R and define a emph {duality-test} which circumvents Kaplansky's counterexamples. We say that M has emph {almost a trivial dual} if there is no homomorphism from M emph {onto} a free R-module of countable (infinite) rank. In the first part of this paper we must strengthen and adjust the new combinatorial principle (called the aleph_n-Black Box) and in the second part we will apply it to find arbitrarily large aleph_n-free R-modules over complete DVD with almost trivial dual. A corresponding result for torsion modules is obtained as well. Also observe, that the existence of such modules can easily be established assuming GCH (see e.g. Eklof-Mekler [EM] on diamonds), so the problem rests on the fact that we want to stay in ordinary set theory.

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