Pcf and Abelian Groups

We deal with some pcf, (posible cofinality theory) investigations mostly motivated by questions in abelian group theory. We concentrate on applications to test problems but we expect the combinatorics will have reasonably wide applications. We almost always answer the test problem which is proving the existence of ℵω-free abelian groups with trivial dual, i.e., with no non-trivial homomorphisms to the integers. More specifically, we prove that " almost always " there are F ⊆ κ λ which are quite free and have a relevant black box. The qualification " almost always " means that except when we have strong restrictions on cardinal arithmetic, in fact those restrictions are " everywhere ". The nicest combinatorial result is probably the so called " Black Box Trichotomy Theorem " proved in ZFC. Also we may replace abelian groups by R-modules, part of our motivation is that in some sense our advantage over earlier results becomes clearer in such context.